So high are the Brexit stakes that divorce talks were always likely to go to the line. After yesterday’s “deadlock”, this appears to be the case. There must surely be an urgent heads of government session, at least involving Britain, France and Germany, to cut a deal on cash and talks on trade. It is time to bring on the grownups.

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But all politics is local, especially Brexit politics. It is impossible for Theresa May to embark on this next stage with a divided cabinet, party and parliament. The kindest analysis of her tactics so far is that she has played cunning. She has kept her Brexit backwoodsmen inside the cabinet tent, and has resolutely refused to seem partisan herself. That has not worked – witness the latest fierce backwoods demand for the chancellor to be sacked. There is now a serious risk of Tory right/Labour left collusion. As Brexit legislation enters parliament, paralysis by amendment will see May fighting on two fronts at once, at home and abroad. This is unsustainable.

All involved in this mess should keep one fact in mind. Britain has agreed to leave the EU, but it has not agreed how. Every opinion poll, every interest group, every sensible bit of evidence is tilted towards “soft” Brexit, towards workable, feasible, sane Brexit. Even polls on immigration specifically among leavers – by YouGov, the Economist and King’s College London – indicate majority acceptance of free movement for EU workers, given restrictions on benefits similar to those operating in many other EU states.

There is no consensus for a “cliff edge” departure from the EU, for customs barriers, trade quotas, passport restrictions and no transition. There is no majority for hard Brexit in parliament, even among Conservative MPs. Enthusiasm for the World Trade Organisation option is limited to a few newspapers and the Tory right. To any bank, business, travel organisation or university, hard Brexit is flat-Earthism.

Some compromise with the EU will have to come. It has already been hinted at over the budget and the European court. But the next “compromise phase” will collapse if May has to be looking over her shoulder every hour of the day in the House of Commons. This will especially be the case should Labour and the Liberal Democrats cynically use Brexit as a manoeuvre to bring her down.

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May’s primary task of leadership is to contrive a parliamentary “coalition of the sane” on Europe for the duration of the talks. It may run against her nature to get the party leaders and their whips to agree on how best to monitor and support the negotiations. But there is a clear public interest, indeed public demand, for this to happen. Britons clearly wish to retain their open market status within Europe. Parliament should reflect that wish, and not let a minority drive negotiations to a crash.

• Simon Jenkins is a Guardian columnist