WOOD, Circuit Judge.

The Supreme Court has held that Section 301 of the Labor Management Relations Act (LMRA), 29 U.S.C. § 185, preempts all state-law claims that require the interpretation of a collective bargaining agreement (CBA) or any other covered labor contract. This rule, which goes under the clumsy name of "complete preemption" (a better term might be "complete displacement"), covers not only obvious disputes over labor contracts, but also any claim masquerading as a state-law claim that nevertheless is deemed "really" to be a claim under a labor contract. These cases are understood to arise under federal law (Section 301 of the LMRA) from their inception. Subject-matter jurisdiction in the federal court thus rests on 28 U.S.C. § 1331, and it makes no difference if the parties are nondiverse or if the plaintiff's complaint is couched in state-law terms.

Cooper B-Line, Inc. contends that this is one such case. Although Philip Crosby, a former Cooper employee, filed a lawsuit in state court that asserted only a state-law claim against the company — the Illinois tort of retaliatory discharge — Cooper contends that the suit is a disguised Section 301 action that arises under federal law. Cooper removed the case to federal court on this basis and now asks us to dismiss it in its entirety because Crosby failed to exhaust his remedies under the CBA. Crosby counters that removal was improper and asks us to return his case to state court. We conclude that this case is not materially different from the one before the Supreme Court in Lingle v. Norge Div. of Magic Chef, Inc., 486 U.S. 399, 108 S.Ct. 1877, 100 L.Ed.2d 410 (1988), where the Court rejected a finding of complete preemption. We do too, so we reverse the district court's judgment on the merits in Cooper's favor and order the case remanded to the state court.

I

Cooper describes itself as "a diversified global manufacturer of electrical components and tools," see COOPER INDUS., http:// www.cooperindustries.com/content/public/ en/company.html (last visited Aug. 5, 2013); it has seven operating divisions, one of which is Cooper B-Line, id. Crosby worked in the "flex tray" department of Cooper's facility in Pinckneyville, Illinois. On July 28, 2010, a portion of Crosby's middle finger was amputated as he was attempting to remove a piece of flex tray metal from a master bundle. Crosby had instructed a co-worker to kick the bundle in order to dislodge the pieces as he removed them. The company frowned on this procedure, precisely because it could lead to serious injury. Crosby sought medical attention and filed a claim for medical and temporary total disability benefits under the Illinois Workers' Compensation Act. 820 ILCS § 305.

Crosby returned to work on September 13, 2010. Following a conversation with management in which Crosby stubbornly maintained that he did not intend to stop using the "kicking method" to remove metal from the master bundles, Cooper suspended him for three days without pay as discipline for using an unsafe work practice. The "Disciplinary Corrective Action Discussion Worksheet" accompanying the suspension further stated that any violation of Cooper's safety policies for the duration of Crosby's employment would result in immediate termination of his job. Dennis Zimmerman, president of Crosby's union (Local 7252, United Steelworkers of America), filed a grievance on Crosby's behalf to protest the suspension.

After returning from his suspension, Crosby was given additional safety training during which, he alleges, Cooper told him about numerous new safety rules and procedures. Crosby then resumed work in the flex tray department, but, within hours, Cooper's plant safety specialist accused him of violating one of the new safety rules by tossing a wooden pallet. Crosby denied that he did so.

Cooper personnel then convened a meeting with Crosby and Zimmerman to discuss the alleged safety violation. At some point, Crosby was asked to leave the meeting, and the Cooper representative told Zimmerman that Crosby would be fired. Zimmerman left the meeting to give Crosby the bad news; he suggested that Crosby ask Cooper to call its action a "permanent layoff with no recall rights," rather than an outright termination. If it did so, Crosby would be able to obtain unemployment benefits and a neutral job reference. Crosby followed Zimmerman's advice and asked Cooper for the more favorable label. Cooper accepted on the condition that Crosby agree to dismiss the grievance filed in response to his three-day suspension. The parties' agreement was memorialized in a "grievance settlement" (drafted by Cooper), which states, in its entirety:

Crosby now says that the entire process leading up to his signing the grievance settlement was a sham. He believes that Cooper's real reason for discharging him had nothing to do with safety violations. Instead, he asserts, it was because he filed a workers' compensation claim after the amputation of his finger. In support of this theory, he points to the timing of his firing, the sudden appearance of new safety rules to which he was subject, several emails from Cooper personnel expressing a desire to fire him or otherwise send a "hard message" immediately following the July 28 accident, and emails from Cooper's Human Resources manager expressing concern that transferring Crosby to a different part of the plant would result in a more costly workers' compensation award.

This view led Crosby to file a suit in the Circuit Court for the Twentieth Judicial Circuit, Perry County, Illinois, in 2011, alleging that he was fired in retaliation for asserting his right to medical and disability payments in violation of the Illinois Workers' Compensation Act. 820 ILCS § 305. Cooper removed the case to the Southern District of Illinois. Because the parties are non-diverse, the only possible basis for subject-matter jurisdiction was the presence of a federal question under 28 U.S.C. § 1331. Cooper argued that Crosby's retaliatory discharge claim was really a claim under his CBA and thus was necessarily covered by Section 301. See, e.g., Caterpillar Inc. v. Williams, 482 U.S. 386, 393-94, 107 S.Ct. 2425, 96 L.Ed.2d 318 (1987). Crosby did not move to remand.

After conducting discovery, Cooper moved for summary judgment. The district court granted the motion, finding that Crosby's retaliatory discharge claim failed on the merits. On appeal, Crosby now contends for the first time that the district court had no subject-matter jurisdiction over the case and asks that it be remanded to state court. For its part, Cooper defends the district court's judgment, but it also argues that the court should not have reached the merits of the case for a different reason. In its view, because the retaliatory discharge claim should have been characterized as a Section 301 claim, it could not be filed unless or until Crosby exhausted all contractual remedies under his CBA before resorting to federal court. Because Crosby did not do so, Cooper argues that this is an independent reason why his suit should have been dismissed. As we explain below, we have no need to reach the latter argument.

II

A

This case turns on a single question: does Section 301 of the LMRA govern the issues that Crosby was attempting to raise in his state-law claim for retaliatory discharge (or, put conventionally, does the LMRA completely preempt Crosby's state-law claim). If the answer is yes, then the case was properly dismissed. If the answer is no, then there is no federal question in the case and the district court lacked jurisdiction to decide it. Although Crosby did not raise subject — matter jurisdiction before the district court, this objection is not subject to waiver. See 28 U.S.C. § 1447(c); Henderson ex rel. Henderson v. Shinseki, ___ U.S.___, 131 S.Ct. 1197, 1202, 179 L.Ed.2d 159 (2011); see also Fed. Sav. & Loan Ins. Corp. v. Quinn, 419 F.2d 1014, 1016 (7th Cir.1969). The federal courts have an independent "obligation at each stage of the proceedings to ensure that [they] have subject matter jurisdiction over the dispute." Ne. Rural Elec. Membership Corp. v. Wabash Valley Power Ass'n, Inc., 707 F.3d 883, 890 (7th Cir.2013). We review questions of subject-matter jurisdiction and the propriety of removal to federal court de novo. Farnik v. Fed. Deposit Ins. Corp., 707 F.3d 717, 721 (7th Cir.2013).

Ordinarily, the basis for federal-question jurisdiction must be apparent from the face of the plaintiff's well-pleaded complaint. See Louisville & Nashville R.R. v. Mottley, 211 U.S. 149, 152, 29 S.Ct. 42, 53 L.Ed. 126 (1908); see also Ne. Rural Elec. Membership, 707 F.3d at 890. The Supreme Court, however, has long recognized an "independent corollary" to the well-pleaded complaint rule that it has dubbed the "complete preemption doctrine." Caterpillar, 482 U.S. at 393, 107 S.Ct. 2425. At its base, this doctrine reflects an understanding of the coverage of the federal law in question. In a small number of areas, "the Court has concluded that the pre-emptive force of a statute is so extraordinary that it converts an ordinary state common-law complaint into one stating a federal claim for purposes of the well-pleaded complaint rule." Id. (internal quotation marks omitted). "Once an area of state law has been completely pre-empted, any claim purportedly based on that pre-empted state law is considered, from its inception, a federal claim, and therefore arises under federal law." Id.

Section 301 of the LMRA provides a federal rule for contract disputes between employers and labor organizations or between different labor organizations. 29 U.S.C. § 185(a). The Court has understood this to be a rule that overrides all possible applicable state law, or, in other words, as a rule of complete preemption. E.g., Franchise Tax Bd. v. Constr. Laborers Vacation Trust, 463 U.S. 1, 23-24, 103 S.Ct. 2841, 77 L.Ed.2d 420 (1983). Given the force of complete preemption, it is not enough for a court to accept a plaintiff at her word when she represents that her case raises no federal issue. Rather, to determine whether a purported state-law claim "really" arises under Section 301, a federal court must "look beyond the face of plaintiff's allegations and the labels used to describe her claims and ... evaluate the substance of plaintiff's claims." Paul v. Kaiser Found. Health Plan, 701 F.3d 514, 519 (6th Cir.2012) (emphasis in original).

Although complete preemption sweeps away many state-law theories and re-classifies them as federal, it has its limits. The Supreme Court has made clear that preemption does not extend to every state law that touches upon rights governed by Section 301 — that is, rights that may have some connection to a CBA. See Lingle, 486 U.S. at 412-13, 108 S.Ct. 1877. Rather, only those state-law claims that require "interpretation" of a CBA are inevitably federal. Douglas v. Am. Info. Techs. Corp., 877 F.2d 565, 569-70 (7th Cir.1989); see also Atchley v. Cable Vision Assocs., 101 F.3d 495, 499 (7th Cir.1996). Put another way, a state-law claim is "completely preempted" only when it is "inextricably intertwined with consideration of the terms of the labor contract." Allis-Chalmers Corp. v. Lueck, 471 U.S. 202, 213, 105 S.Ct. 1904, 85 L.Ed.2d 206 (1985). Factual overlap between a state-law claim and a claim one could assert under a CBA is not necessarily sufficient. Douglas, 877 F.2d at 570. And the general rule that a federal defense does not suffice to support federal subject-matter jurisdiction, e.g., Rivet v. Regions Bank of La., 522 U.S. 470, 475, 118 S.Ct. 921, 139 L.Ed.2d 912 (1998), retains its force, even when complete preemption is at issue, Caterpillar, 482 U.S. at 398-99, 107 S.Ct. 2425. As the Court put it in Caterpillar:

Id. (emphasis in original).

B

The state-law claim at issue in this case is for retaliatory discharge in violation of the Illinois Workers' Compensation Act. 820 ILCS § 305. Although the Act itself does not recognize a cause of action for employees who are fired in retaliation for exercising their rights under the statute, Illinois courts have recognized an implied private action, to "ensure that the sound public policy underlying the [Act] c[an] not be frustrated by the actions of an employer." Hinthorn v. Roland's of Bloomington, Inc., 119 Ill.2d 526, 116 Ill.Dec. 694, 519 N.E.2d 909, 911 (1988). A claim for retaliatory discharge requires a plaintiff to show that "(1) he was discharged or threatened with discharge and (2) the employer's motive in discharging or threatening to discharge him was to deter him from exercising his rights under the Act or to interfere with his exercise of those rights." Lingle, 486 U.S. at 407, 108 S.Ct. 1877 (quoting Horton v. Miller Chem. Co., 776 F.2d 1351, 1356 (7th Cir.1985)); see also Hinthorn, 116 Ill.Dec. 694, 519 N.E.2d at 911. None of these elements immediately calls to mind a labor contract. It is thus not surprising that when the Supreme Court addressed whether complete preemption applied to this very type of claim in Lingle, it ruled against such a finding. 486 U.S. at 407-08, 108 S.Ct. 1877. As the Court pointed out, the elements of a retaliatory discharge claim are "purely factual" and "pertain[] to the conduct of the employee and the conduct and motivation of the employer." Id. at 407, 108 S.Ct. 1877. It is not ordinarily necessary to consult the terms of a labor contract to know whether an employee was discharged, nor would the contract be required to explain why the discharge occurred. While Lingle does not hold that every claim for retaliatory discharge will escape Section 301 preemption (because the facts do matter), it suggests that preemption will be rare.

Cooper acknowledges Lingle, but it contends that this case is different, both because the grievance settlement Crosby signed on his departure from the company was reached under the auspices of the CBA and because it sees the interpretation of this settlement as critical to Crosby's state-law claim. It argues that before Crosby may invoke the protection of the Workers' Compensation Act, he must show that he was discharged. The grievance settlement, however, says that Crosby agreed to a "voluntary separation from the company [that] shall be considered a permanent layoff without recall rights." Cooper argues that the reference to a "voluntary separation" in the settlement prohibits Crosby from claiming he was discharged unless he first attacks the settlement itself. Since attacking the settlement would require the court to interpret and evaluate the CBA's dispute resolution procedures, Cooper reasons that this case, at its core, involves interpretation of a labor contract and therefore falls under Section 301.

We have no quarrel with the proposition that Crosby must prove discharge to succeed on the merits of his claim, see Hinthorn, 116 Ill.Dec. 694, 519 N.E.2d at 911, but we are not persuaded that he can do so only by attacking the grievance settlement. The settlement is not nearly as clear in characterizing Crosby's departure from the company as "voluntary" as Cooper portrays it. The settlement refers to a voluntary separation, but it then explains that the separation is a "permanent layoff without recall rights." That does not sound voluntary to us; it sounds as if Cooper has firmly shut the door behind Crosby. The message we take from the settlement is that Crosby is leaving Cooper and that he does not have the option to come back.

The significance of the grievance settlement wanes further when we consider that Illinois law does not attach dispositive weight to the label an employer uses to describe an employee's departure in determining whether that employee was discharged. In Hinthorn, the plaintiff was ordered to sign a so-called "voluntary resignation" form after being told that she was going to be fired for having cost her employer too much money for workers' compensation claims. Id., 116 Ill.Dec. 694, 519 N.E.2d at 910-11. Her employer told her that she would nevertheless be given an opportunity to "resign" and thereby "save face" and keep a clean resume. Id., 116 Ill.Dec. 694, 519 N.E.2d at 912. The Illinois Supreme Court rejected the employer's attempt to rely on the "voluntary resignation" label in arguing that it did not discharge the plaintiff, finding that the circumstances under which the plaintiff signed the form provided ample evidence of discharge, regardless of what the employer chose to call it. Id. Under Hinthorn, Crosby too is not bound by labels, but instead is entitled to rely on the circumstances leading up to his signing the grievance settlement to show that he was discharged. This evidence has nothing to do with interpreting either the CBA or the settlement agreement.

Instead, the grievance settlement takes Crosby's discharge as given. It governs the terms of that discharge, but it does not undermine the basic fact of the action. The grievance settlement says nothing at all about the central question in a retaliatory discharge case, which is why the employee was discharged. Crosby aims to show that Cooper's desire to punish him for seeking workers' compensation led it to place him (improperly) in the position of having to sign the settlement. In short, interpretation of the settlement is unnecessary to the resolution of Crosby's claim.

Crosby has no wish to repudiate the grievance settlement, and even if he succeeds in this lawsuit, any such success would not deprive Cooper of any benefit it bargained for in the settlement. Crosby is not seeking reinstatement to his former position, nor does he wish to change any other term of the settlement. Crosby wants money damages in compensation for Cooper's alleged violation of Illinois law and public policy, and nothing in the settlement precludes such relief. To the contrary, the settlement says that it "is without prejudice to the contractual rights of either party and shall not constitute a precedent with respect to any like or related matter in the future." If this means anything, it must indicate that both parties are reserving their rights under other applicable laws.

None of this is to say that Cooper is barred from raising preemption as a defense on the merits in state court. See Caterpillar, 482 U.S. at 398, 107 S.Ct. 2425 (claim that is not subject to federal jurisdiction under the complete preemption doctrine may nevertheless be proven preempted when the state court reaches the merits); Smith v. Colgate-Palmolive Co., 943 F.2d 764, 770 (7th Cir.1991) (same). We conclude only that Crosby's case is not within the territory that Congress carved out to be governed exclusively by federal rules in Section 301. Without Section 301 "complete preemption," there is no basis for federal subject-matter jurisdiction over this case. Removal was therefore improper, and we REVERSE and REMAND to the district court with instructions to return this case to state court. Needless to say, we express no view on the underlying merits.