On Thursday, May appeared to secure little in terms of specific concessions, as European Union leaders held firm to their demand for an ironclad guarantee that Ireland’s border with Northern Ireland would remain open, a sticking point in the negotiations.

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In a joint statement with European Commission President Jean-Claude Juncker, May described her talks as “robust but constructive” — diplomatic lingo for saying they made little headway. Juncker, speaking on behalf of the European Union, said that the bloc would be happy to make minor changes to the nonbinding promises it was offering Britain but that it would not budge on the core issues.

“Both sides have made significant concessions to arrive at a deal,” Juncker said in the statement.

The two sides agreed to restart discussions about aspirations for their post-divorce relationship — a relatively minor step that diplomats nevertheless painted as a significant change, in a measure of the low expectations for the discussions.

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Tusk’s comments a day earlier contained some spice, but he was articulating widely held frustration across the European Union that British lawmakers appear unable to settle on any single plan that meets their own red lines.

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May told reporters after the meetings that Tusk’s language “caused widespread dismay in the United Kingdom. The point I made to him was that we should both be working to ensure that we can deliver a closer relationship between the United Kingdom and the European Union in the future, and that’s what he should be focusing on.”

E.U. leaders have repeatedly asked May to propose a concrete plan that would usher Britain out of the bloc without imposing a hard border between Ireland and Northern Ireland, which could risk inflaming the dormant conflict there.

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The plan that E.U. and British negotiators dreamed up together to keep the border open — an insurance policy known as the backstop that could risk leaving Britain trapped half inside, half outside the European Union — has been toxic in the British Parliament.

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May and Brexiteers within her party have demanded a way to avoid getting stuck permanently halfway out the door. But E.U. policymakers say they cannot countenance anything that would jeopardize the Irish border.

Without a deal in place or an extension of the talks, Britain is due to crash out of the European Union without a safety net March 29. An uncontrolled departure could set off economic and social chaos on both sides of the English Channel, and worried E.U. leaders are stepping up preparations to deal with the fallout.

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One element that had a small potential to change the long-term dynamic of the discussions was a letter late Wednesday from British Labour Party leader Jeremy Corbyn, who offered opposition support to the divorce deal if May can guarantee a close, long-term relationship with the European Union.

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Corbyn wants Britain to remain in a customs union, probably meaning that it would remain closely aligned with E.U. rules and regulations and could be hard-pressed to secure independent free trade deals, as many Brexit advocates demand. Remaining in a customs union would allow for an open border between Ireland and Northern Ireland.

Corbyn’s proposal probably would be largely acceptable to E.U. leaders. But if May got her Brexit deal passed with help from Labour but not from her own hard-line Conservative backbenchers, it would split her beloved Tory party.

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“We believe that a close economic relationship along these lines would make it far less likely that any backstop arrangements would ever be needed,” Corbyn wrote in his letter.

One E.U. lawmaker who met May on Thursday welcomed the proposal.

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“I welcome @jeremycorbyn letter making a cross-party approach for the first time possible,” the European Parliament’s Brexit coordinator, Guy Verhofstadt, wrote on Twitter. “From the hell we’re in today, there is at last hope of a heavenly solution even if it won’t be Paradise.”