A casual observer seeing Theresa May and Jeremy Corbyn exchange rhetorical blows over Brexit in the Commons today might imagine that they stand on opposite sides of this totemic issue. But veterans of the debate know they are equal in determination that Britain should leave the EU. They are separated from consensus by small technical matters that can, for theatrical effect, be inflated into irreconcilable differences. That process is testimony to the hold that the two-party system exerts over English politics and to much of what is wrong with it.

Brexit keeps promising to break party lines and then failing to do so. That is partly because pro-Europeanism and the British left stand on overlapping cultural ground. The remain proposition goes well beyond attachment to EU membership: it is driven by revulsion at the nativist spirit of Brexit’s marauding generals – Nigel Farage, Boris Johnson, Jacob Rees-Mogg – and shock that they have hijacked Britain’s political identity. But that dismay is not exclusive to Labour. It is felt by liberal Tories and swing voters who just pine for competent, rational government.

The result of the last general election was confusing, in that sense. Corbyn’s strength on the stump accounts for some of Labour’s performance. But so does May’s self-destructive campaign, reinforcing caricatures of Tory nastiness and repelling anyone with doubts about Brexit. On 8 June 2017 three forces combined to Labour’s benefit: Corbynism, vaguer anti-Toryism and remain – in what proportions no one knows. The fault lines were then concealed by an outbreak of loyalty among Labour MPs. Those who had expected voters to despise Corbyn’s candidacy were chastened. For various reasons (some genuine reappraisal of the leader’s qualities and an absence of better ideas) the moderates mostly fell into line. Many found sanctuary in the pro-European cause, which brought reconciliation with the rank and file membership. MPs who had been out in the cold because of their hostility to Corbyn took a warm bath in grassroots affinity for remain.

The internal opposition was dormant, not extinct. Corbyn’s mealy-mouthed response to the Salisbury poisonings, marked by reluctance to blame the Kremlin, drove some MPs to distraction. His refusal to show any regret for a career spent rubbing shoulders with antisemites has pushed others to despair. But the European question has kept many dissidents from quitting. They thought they had a better chance of influencing the Brexit outcome from inside the party. That conviction has crumbled in recent weeks. Corbyn’s Euroscepticism is on display more clearly than at any time since his days as a backbencher, when he denounced the EU as a sinister imperial project. Old videos of him railing against the Brussels enemy as a “military Frankenstein” are now gleefully shared by Ukippers online. The Labour leader’s constant reluctance to back a second referendum was for months excused as tactical caution. Increasingly, it looks like sabotage.

How many voters would identify with a movement born in simultaneous aversion to Corbyn and Brexit? No one can be sure

Labour pro-Europeans have been taken for a ride. Their Tory equivalents are now grasping the scale of their symmetrical defeat. Moderate Conservatives once predicted that Brexit bluster would be dispelled by diplomatic reality and that Commons arithmetic would impose a sensible settlement on May. There is still time for that to happen, but not much. Away from Westminster, the party is in an uncompromising mood. Earlier this week, Grantham Conservatives initiated the process to deselect Nick Boles, their local MP. His offence was to say he would renounce the Tory whip sooner than endorse a no-deal Brexit. A few days earlier, Labour’s MP for Liverpool Wavertree, Luciana Berger, had faced charges of disloyalty to Corbyn. A no-confidence motion against her was withdrawn only when noxious social media posts from within the constituency party made it impossible to deny that Berger was targeted in part because she is Jewish.

These episodes are symptomatic of a purifying mania that will shrink the two broad churches of English politics into intolerant sects. One quivers in anticipation of Brexit rapture; the other hunts blasphemies against the supreme leader. Many Labour MPs see this process as irreversible. Corbyn commands the members’ faith. When he goes, his successor will pledge to honour his legacy. A similar realisation is dawning on moderate Tories. A successful candidate to replace May must seek endorsement from the Europhobic backbench commissars.

It is certain, on those trajectories, that MPs will leave their current parties. But to what end? Obstacles to a new venture are familiar. The electoral system suffocates startups. A gang of refuseniks from left and right would disagree on issues that Brexit has obscured, starting with tax-and-spend policies. There is already a pro-EU centre-party, the Liberal Democrats, which is not surfing any vast anti-Brexit backlashes. Schismatic Labour MPs say Vince Cable’s tribe would fold into a fresher-looking insurgency, but even then the target audience is uncertain. Labour and Tories marshall diverse voter coalitions. A post-remain newcomer would energise middle-class metropolitan graduates, which is a launchpad, not a wide base. Miserable Labour and Tory MPs know all of those problems, but one factor is overcoming fear of breaking away; abhorrence at the thought of staying where they are.

The standard accusation against “centrist” politics is that it is all tactical calibration with no ethical compass. It is scorned by radical left and right alike as a cavilling creed for unprincipled careerists. But the careerist route doesn’t run through the centre any more. The two main parties still control the best jobs in British politics. The greasy poles to climb are planted in the camps of Corbynism and Europhobia. In that sense, the strongest argument against a new party – that it would be pointless, and would fail – is the best argument for doing it anyway. A futile gesture at least looks motivated by something better than cynical ambition. Showing no fear of failure is one way the enterprise would look principled.

How many voters in the country would identify with a movement born in simultaneous aversion to Corbyn and Brexit? Could it be viable in an election? No one can be sure. The only way for rebellious MPs to find out is to admit that they have no better options, summon the courage, and try.

• Rafael Behr is a Guardian columnist