When Theresa May calls the cabinet to order on Wednesday morning, amid the chintz of Chequers, the roses will be past their best and the lawn covered with dew. After Britain’s mad summer, the May administration will convene in Buckinghamshire finally to face the chill autumn reality: Britain voted for Brexit without a plan and the Europeans intend to shaft us.

They may not have actually used the word “shaft” when they met on the Italian aircraft carrier Giuseppe Garibaldi last week, but German chancellor Angela Merkel, French president François Hollande and Italian prime minister Matteo Renzi know they hold all the strongest cards. French civil servants are trained at the grandes écoles, like diplomatic special forces, to spot and ruthlessly attack the weakness of negotiating partners. And May’s administration goes into this critical negotiating process with an obvious weakness. It does not know whether it wants to remain in the single market.

Chancellor Philip Hammond is said to want “partial” single market access, to keep the City’s passporting arrangements into the eurozone. If, in order to get it, Britain has to soft-pedal on immigration controls, that would be no disaster for the Treasury, whose growth projections in this year’s budget relied on the impact of a million EU migrants over the next five years.

Brexit minister David Davis and international trade secretary Liam Fox are champing at the bit to begin the article 50 process – naturally they have drawn up rival timelines – but have so far failed to produce any kind of blueprint for their preferred outcome, which is to quit the single market and end free movement. This leaves May struggling to assert control over the process of even coming to a negotiating strategy – hence Wednesday’s away day – and in severe danger of floundering once she has to deliver her Brexit plan to the other EU heads of government.

It is a real, serious and material split in the government of a major country. And it is backed by rival forces in society. Those for whom Brexit became a religion in the spring of 2016 do not care about the niceties of the European Free Trade Association and the European Economic Area. They voted to leave all of it and to “take control” of migration. But the economic elite of Britain, which has the strongest voice both in the Conservative high echelons and in the civil service, simply does not want Brexit. Above all it wants to maintain market access for the City, and for the major global service firms headquartered in London. For them, it is logical to hope that Europe stonewalls all May’s requests for flexible market access, and that – by the end of the process, and with the economy suffering – the public will be ready to accept staying in the EU, with some minor variations on migration.

Labour, too, is racked by the same dilemma. John McDonnell’s “red lines” on Europe do not, and cannot, include keeping free movement, but do include keeping the City’s financial passport and single market “access”. For Jeremy Corbyn the additional problem is that the European left wants Britain out of the EU pronto, before it can negotiate any dilution of the social chapter.

Both main parties, then, are trapped between what is possible and what the British people voted for. Owen Smith may be wowing his rallies with talk of a second referendum, but the party leaderships are beginning to realise this will only be sorted out in parliament. It will be complex, messy and reputations will be minced. The only democratic way of doing it is to say – if necessary with May, Corbyn and Nicola Sturgeon standing shoulder to shoulder somewhere symbolic – that Britain’s aim is to remain in the single market.

Until the delusion of the giant flounce-out is killed off – and until Liam Fox and David Davies are told to stop dreaming about it – those negotiating with Britain from the other side will have a massive incentive to force us into it. If they think it is our secret desire, or our plan B, for Britain to quit the single market cleanly, Europe will not take seriously any UK government demands for variations to single market rules, designed to keep us on the inside the market, though outside the EU itself.

May should seek cabinet agreement for a baseline request: to remain as part of the EEA, with a humane, time-limited restriction on free movement as Britain’s one demand, and then get on with it. She should make a statement to the commons that rules out leaving the single market. Any confusion that is allowed to fester over this will corrode her own authority as the Europeans sensibly deploy every trick in the book – a book going back to the era when Prince Metternich ran rings round Britain in post-Napoleonic Europe – to make her squirm.

There will be uproar in Ukipland – but there always is. Not only is there no clear mandate for leaving the single market, but the negotiating positions of our major partners are now clear. They have not said “go ahead, pick and mix” from the EU goodie-bag: they have said the single market plus free movement or get lost. The British negotiators’ job is to prise that position apart – not rubberstamp it, which is what Fox, Davies and other hardcore Brexiters want.

May wants to serve a full term. But both logic and principle dictate that were she to give in to the “clean break” brigade within the cabinet, she would have to schedule an election and fight for a mandate to lead Britain into this particularly stupid form of economic suicide. If so, whoever leads Labour should be salivating at the prospect. The Brexit moment caused many in the City to question the long-term direction of centre-right politics. By finessing May into power without a vote, and surrounding her with the grey men and women of provincial Toryism, the grandees of the party restored the impression that they could – just about – hold the line for the City, the big law firms, the service giants and big pharma.

But if Tories want to go into an election promising to destroy 30 years of European financial integration in order to assuage the xenophobes and climate deniers of Ukip, that would be a different prospect.

For the opposition parties, if they can recover their nerve, any confusion coming out of Wednesday’s cabinet – still better any open revolt – will be a big opportunity. They should state, simply, that they will seek a vote in parliament to veto any Brexit strategy that tries to remove Britain from the single market. That principle – which needs no further elaboration since these are opposition parties, not a parallel government – could become the rock upon which Theresa May’s soaraway ratings are broken, and her majority ground down.