Sticking to their own script, the other 27 states gave May until the next regular summit in eight weeks to improve an offer officials estimate at about €20bn if she wants them to start discussing future trade ties. Miss that deadline and, the EU says, time will be running out for any deal.

In essence, the 27 need to trust May that Britain will pay much more than is on the table but understand her difficulties in naming a figure by December, which could spark revolt at home and derail the process. In return, they seem likely to let slip more hints of what kind of future relationship she might secure.

The EU position is “solid”, Italian Prime Minister Paolo Gentiloni told reporters after the talks, but also “flexible”.

It was unrealistic, he said, to expect May to sell a precise demand for money in December. Equally, she could not expect the EU to negotiate a future trade deal without knowing roughly what outstanding items Britain would pay for. But defining those items could, Gentiloni said, be done “in the most politically manageable way possible for our British friends”.

Even the roughest of definitions will let commentators work out numbers. But as a senior diplomat from another major EU power put it: “We don’t want to go public with a bald figure. If it’s on the front page of The Sun, the whole process is dead.”

“Reports of the deadlock ... have been exaggerated,” summit chair Donald Tusk concluded, adding that discussions with May had finally succeeded in “establishing trust and goodwill”.

Tusk, a former Polish premier, echoed the EU mantra of unity among the 27 and full backing for European Commission negotiator Michel Barnier, whose team of technical experts are handling the talks with London. But he also acknowledged a nuance in his role in steering the high-level political imperatives of the leaders.

That, Tusk said, meant he would be a “positive motivator” in the coming weeks to create “a more positive narrative” than what some British politicians call EU “blackmail” demands.

Anxious not to worsen May’s troubles at home, fellow leaders made an effort to demonstrate goodwill. German Chancellor Angela Merkel and French President Emmanuel Macron take a hard line on “sequencing” divorce talks before trade, but they huddled with May in amicable conversation for television cameras.

The EU communique acknowledged progress on key issues, which also comprise expatriate rights and the Irish border, and instructed Barnier to begin internal EU preparations for the talks May wants on a two-year transition period after Brexit.

Tusk said EU preparations would “take account of proposals presented” by “our British friends” -- an indication that, even without direct talks, May might be able to show her domestic audience an outline of how the EU sees future ties.

Some were disappointed that she repeated Britain’s rejection of the legal basis of much of the EU’s demands, notably that it pay for EU commitments lasting beyond 2020. Macron, for one, said they were “not even half way” toward an agreement on money.

But others detected a more positive tone. Luxembourg Prime Minister Xavier Bettel said May had indicated that Britain was at least “analyzing” what other parts of the bill it might pay. Merkel said she had “no doubt” a good agreement was possible.

Yet there are few illusions that the Brexit plot will play out smoothly. “We’ve always predicted a drama for autumn,” one EU diplomat said. “We haven’t seen it yet. So maybe November.”

UK's divorce bill figure will come in final Brexit deal: trade minister