Imagination, agility, empathy, diplomacy – all the qualities of an effective negotiator that Theresa May lacks. She was unsuited to the task of getting a good deal in Brussels and now looks incapable of steering a bad deal through parliament. But the prime minister is not without skill. She has an exceptional ability to drain the drama from a crisis, to eke dullness out of an emergency. It is practically a superpower. Only May could make a live broadcast in a drastic breaking-news situation feel as missable as a daytime repeat. As one former Downing Street aide tells it, May has one political technique that she applies to every situation: “She just grinds you down.”

It seemed inconceivable that May would try to rehabilitate her Brexit plan after it was rejected in the biggest Commons defeat for a sitting government in modern times. MPs on all sides cheered the deal’s annihilation, believing that the scale of the shock would jolt the prime minister into a change of course. It has not. May returned to parliament this week looking like an earthquake survivor unable to take in the magnitude of the devastation, pottering numbly around a ruined house, rehanging pictures on half-collapsed walls, perching ornaments on the rubble that was once a mantelpiece.

May’s plan B is plan A with a lick of paint. Her pitch to MPs has two components. First, a tweak to the backstop that is designed to insure against the return of a visible border in Northern Ireland, which hard Brexiters despise as a trapdoor to a perpetual customs union with the EU. Second, May promises deeper parliamentary involvement in shaping the longer-term relationship with Brussels. But no backstop concession has yet been requested in Brussels and none yet looks available. And no one who has met May believes her capable of sustained bipartisan collaboration.

May’s attachment to the Conservative party is intense, even for a Tory MP. She put in eight years as a local councillor before seeking a parliamentary seat. She belongs to the grassroots culture of the party: the pavement-pounding, envelope-stuffing rank-and-file. She says she wants to deliver a Brexit for the whole country but she cannot conceive of doing it on terms that would break Eurosceptic Tory hearts.

That is why her red lines are undiluted. May heard conflicting counsel in response to last week’s events. Gavin Barwell, her chief of staff, favoured deals with the moderate mainstream in parliament, fishing for support in the pool of Labour MPs who would choose a soft Brexit over the trauma of another referendum. Julian Smith, the chief whip, and Robbie Gibb, Downing Street director of communications, steered the other way: redouble efforts to bring the DUP on board in the hope many Tory Brexiters would follow.

That might not be as fanciful as it sounds. The mobilisation of remainers in parliament to thwart a no-deal scenario has made hardliners nervous. Given the backlog of legislation to enact even a negotiated exit, some extension of the article 50 window looks inevitable. Once the totemic date of 29 March has passed without Brexit having happened, the prospect of it never happening feels more realistic. Influential Tory backbenchers are whispering that there would be “movement” if something (anything!) could be done about the backstop. But there is no mood in Brussels for feeding morsels to the wild Tory beast that always just comes back for more.

May’s other route – the coalition of soft Brexiters – is no easier. The opening bid for compromise would be a permanent customs union. Labour says it wants one and enough Tories could stomach one for a parliamentary majority to look feasible. But only if tribal party allegiances are ignored. A Brexit deal that was carried by scores of Labour MPs, tacitly or explicitly backed by Jeremy Corbyn, would rip the Conservative party in half. And Corbyn has no interest in taking ownership of Brexit when a majority of his party’s members want the whole thing scrapped. A transparent objective in the opposition leader’s office is to get through the current crisis without leaving fingerprints on any unpopular resolution. Labour’s official line is for things to be “on the table” and for “options to be considered” in parliament. The moment of actually picking one of those options will be deferred for as long as possible. Corbyn is relaxed about Brexit, but it would suit him if it were a wicked “Tory Brexit” that he can denounce.

Meanwhile, pro-European Conservatives underestimate the gravitational pull of Labour tribalism even on MPs who hate their leader. Pragmatic Conservatives cannot fathom why centre-left MPs who are probably going to be purged as “Blairite” heretics wouldn’t make common cause with like-minded folk on the centre-right whose party has also been captured by maniacs. Tory centrists are confused because they have no emotional equivalent to the “Labour-till-I-die” creed that prevents anti-Corbyn defections. Theresa May probably understands it better because she loves her party the way Labour MPs love theirs.

Brexit has generated cross-party dialogue, even friendships. There have been joint amendments and coordinated rebellions. But it is mostly on the level of procedure and agreement on what shouldn’t happen. Ruling out no deal is the foothills of collaboration. Settling on what should happen is a higher mountain to climb. Dissolving party lines for long enough to pass the necessary legislation goes higher still.

The referendum wreaked havoc with traditional political identities outside Westminster. Many voters now badge themselves as leavers and remainers more readily than they would sport a blue or red rosette. But in parliament the English party duopoly has proved formidably resilient. Tribal impulses run deep. There are few rewards for fraternising with the enemy.

May and Corbyn are set in their ways and alike in craving approval from their party’s least flexible supporters. Their interests are also perversely aligned on Brexit. When May’s deal was rejected, she offered cross-party talks knowing there was no prospect of agreement with the Labour leader. It suited her that he refused the invitation. She likes to have Corbyn lurking on the sidelines, spooking Conservatives with the thought of radical socialist government. It suits Corbyn for May to carry on grinding away as she has always done, painting Tory colours on the Brexit mess from top to bottom. And the deeper we go into this crisis, the louder each side will accuse the other of putting party before country. Both will be right.

• Rafael Behr is a Guardian columnist