Prime minister’s questions, when the House of Commons works itself into an adversarial frenzy, is not a forum made for non-partisan collaboration. So it was not surprising when, on Wednesday, a request that Theresa May convene cross-party hearings on the crisis facing the National Health Service had a mixed response.

Mrs May answered Norman Lamb, the Liberal Democrat former health minister, with chilly civility. A handful of Labour MPs jeered. Mr Lamb, they say, is disqualified from caring about the NHS because he worked with the Tories in coalition. There is nothing unusual about such tribalism in Westminster and the health service always animates deep passions. Moments earlier, Jeremy Corbyn had exposed Mrs May’s lack of a coherent plan to help struggling hospitals. Labour senses that the government is vulnerable on the NHS and is not about to surrender the advantage by signing up to a truce.

If Mrs May were seriously amenable to Mr Lamb’s proposal (and there is scant sign that she is), it would be as a tactical opportunity, drawing the sting from an issue on which she would otherwise be perpetually defensive. But she doesn’t want to inflate the status of the Lib Dems by granting them any convening power or limit her own room for manoeuvre.

Such calculations are unavoidable in politics, even if most voters consider them petty. Only in great national emergencies do rivals set aside visceral mistrust and historical grievance. The challenge facing health and social care services is reaching that scale. Mr Lamb has already recruited a number of Tory and Labour MPs, including select committee chairs and former health secretaries, to his putative convention, which is also endorsed by many medical professional bodies and charities. Away from the pugilism of the Commons chamber, the need for a more considered, evidence-led approach is almost universally recognised.

The potential outline of an ideological compromise is also clear. Labour will have to concede that problems in the NHS are not simply a manifestation of wicked Tory austerity – that an ageing population and the rising cost of treating complex chronic illness makes the current structures hard to sustain even with increased funding. And the Tories must concede that funding is a problem and stop pretending that adequate resources are available. All sides will then have to look at ways to increase resources in ways that garner public consent – hypothecated taxes, for example. And all sides will have to think creatively about reducing demand on the system – serious investment in preventative public health measures, for example, perhaps co-financed by the private sector.

Currently, the gap is unbridgeable. Mr Corbyn needs the NHS crisis as his best available stick for beating the government. Labour shortly intends to defend a byelection in Copeland, Cumbria, entirely on that basis. Meanwhile, Theresa May lacks the imagination to do anything other than deny the problem or try to shift the blame. Eventually, the sheer scale of the crisis must overcome the impasse, forcing politicians to collaborate in search of solutions. The alternative is endless ideological trench warfare without progress or original thinking, leading to a catastrophic decline in vital services. That is an outcome no one in parliament would welcome, and yet sadly it is a scenario in which too many already collude.