Theresa May had set Britain on a course for a hard Brexit, prioritising sovereignty at the expense of close economic ties. Nevertheless, most EU governments had hoped she would win big on 8 June, so that she would be strong enough to face down the Tory right in pushing through painful compromises. They now face a prime minister whose authority is crumbling. Yet the general election makes the prospect of a softer Brexit plausible.

May’s instincts are probably to keep pushing for the hard Brexit that her right wing desires. But there is no parliamentary majority for a hard Brexit. Just a few pro-EU Tories could join opposition MPs to defeat May. If she wants to pass the Brexit deal – and the many Brexit-related laws that are required – she will have to collaborate with Labour and other opposition MPs.

Such a volte-face would be uncharacteristic of May. But if she doesn’t reinvent herself as a soft Brexiter, it is hard to see how she can stay in office. And if she falls, her successor will find that survival means working with the opposition to achieve a softer version of Brexit.

Business leaders sense the chance for a change of direction and are preparing to speak out for a softer Brexit. That is unlikely to mean staying in the single market, since the EU would insist on free movement of labour – a price many Conservative and Labour MPs would not want to pay.

But a softer Brexit could mean introducing only modest curbs on free movement, staying in EU regulatory agencies and avoiding a dogmatic rejection of any role for the European court of justice (ECJ). It could even mean maintaining the customs union. There would then be no need for controls and perhaps queues on the EU-UK border – or for customs posts between the north and south of Ireland. But the UK would have to adopt EU tariffs and could not negotiate its own free trade agreements with countries outside the EU. Staying in the customs union would madden the Tory right as much as it would please businesses.

The shape of Brexit, of course, does not depend only on the UK. EU leaders want a deal, but believe they can insist on their terms, since no deal would damage the UK far more than the continent. They would be happy if the UK sought a softer Brexit, which would be less disruptive for their economies. But they will stick to their principles: the single market must include free movement of people and the jurisdiction of the ECJ; Britain must not “cherry-pick” parts of the market, lest it undermine the EU’s institutional and legal coherence; and life outside the EU must be visibly less agreeable than membership. The EU will make it clear that if you want more economic integration, you must give up sovereignty.

But the EU27 and the Brussels institutions care about more than the substance of the British position. In recent months they have grown increasingly frustrated with May, her team and the key Whitehall ministries, accusing them of being ill-prepared, lacking expertise and making too many unrealistic demands. They are surprised that May has put little effort into building relationships with other European leaders.

The 27 hope for a softer and more realistic British approach – whether they have to talk to May Mark 2 or a new prime minister. If the British oblige, the EU could scale back some of its demands, for example on money. And then a deal would become more likely.

Charles Grant is director of the Centre for European Reform