Theresa May looked tense and unsmiling as she arrived at the Berlaymont in Brussels for the briefest possible photo opportunity with Jean-Claude Juncker before their hour-long meeting.

Both sides had played down expectations about the prime minister’s visit even before Donald Tusk eschatological musings about Brexiteers stirred up a fizz of phoney outrage in London.

At the end of a “robust but constructive” discussion, Juncker and May agreed to meet again before the end of February, and that their officials should meet next week to see if “a way through can be found”.

Yet there was no attempt to disguise the fundamental disagreement between them as May said she needed a legally binding change to the Northern Ireland backstop and Juncker said the EU would not reopen the withdrawal agreement.

The prime minister’s next meeting, with the European Parliament’s president Antonio Tajani and Brexit co-ordinator Guy Verhofstadt, was no less awkward. But she confirmed that, as she said in Belfast this week, she was not seeking to replace the backstop but only to change it to ensure that Britain could not be trapped indefinitely within it.

Privately her aides have told officials in Brussels that the so-called “Malthouse compromise” which calls for the backstop to be replaced with “alternative measures” is a non-starter. Yet ministers and civil servants in London will continue to go through the motions of exploring these alternative measures to appease Conservative Brexiteers ahead of next Thursday’s Brexit debate in the House of Commons.

Five demands

Jeremy Corbyn’s letter to the prime minister has encouraged EU officials to believe that a broadly-based majority can be assembled at Westminster behind a softer Brexit.

Corbyn’s five demands received their most enthusiastic welcome at Westminster on Thursday from advocates of the “Norway plus” model of customs union and single market membership.

The Labour leader’s letter draws back from such a deep level of integration, and some of his proposals are impossible for the EU to accept. However, they would set British negotiators on a path that would allow the backstop to fall away as checks on goods travelling between North and South and between Britain and Ireland would be unnecessary.

May’s next step depends on the fate next week of Yvette Cooper’s amendment that would allow MPs to oblige the government to postpone Brexit rather than leave the EU without a deal. If the amendment passes the prime minister will be driven towards a cross-party solution to the Brexit impasse. If it fails she is likely to cling to her current strategy of seeking a majority based on Conservative and DUP votes, with potentially catastrophic consequences.