Having rejected the exit deal May struck with the European Union, the British parliament on Tuesday voted to send May to Brussels to remove the so-called backstop clause.
While May saw this as an opportunity to prevent the United Kingdom from leaving the EU without a deal, European officials on Wednesday insisted there was no room to rewrite the negotiated deal.
"The Withdrawal Agreement is not open for renegotiation," said European Council President Donald Tusk. "Yesterday, we found out what the UK doesn't want. But we still don't know what the UK does want."
Other frustrated EU officials, including Brexit negotiator Michel Barnier, insisted that the remaining 27 EU members were united and determined not to abandon the backstop they believe is key to maintaining peace on the border.
And he warned that the British vote had only "increased the risk of a disorderly withdrawal" and Northern Ireland "slipping back into darker times past".
The Irish backstop clause is an insurance policy which would guarantee no hard border is placed on the island of Ireland in the event that post-Brexit trade negotiations between the UK and the bloc proved unsuccessful.
Under the terms of the withdrawal agreement, the whole of the UK would remain in a customs union in relation to trade in goods with the EU "unless and until" the bloc agreed there was no prospect of a return to a hard border, while Northern Ireland would also conform to some rules of the European single market.
UK legislators critical of the clause say it threatens the integrity of the the UK's borders and could even lead to the UK staying within the EU customs union permanently.
Critics have argued for the inclusion of a mechanism to allow either side to withdraw from the backstop or a limit to how long it could last.
On Wednesday, May spoke to Tusk and Ireland's Prime Minister Leo Varadkar while also meeting Labour party leader Jeremy Corbyn in an attempt to find elusive cross-party unity on Brexit.
May conceded that her government hadn't settled on a way to replace the backstop, telling MPs that "there are a number of proposals for how that could be done".
She added that measures under consideration included a unilateral exit mechanism from the backstop for Britain, a time limit to the backstop and "mutual recognition and trusted trader schemes".
Varadkar said he would not accept May's plans to rework the so-called Irish "backstop", adding that it needed to be legally robust.
"The Taoiseach set out once again the unchanged Irish and EU position on the Withdrawal Agreement and the backstop, noting that the latest developments had reinforced the need for a backstop which is legally robust and workable in practice," a spokesman for Irish government said after the two leaders spoke by phone.
Corbyn said he "set out the Labour case for a comprehensive customs union with the European Union" during talks with May.
He called the talks "serious" but accused the government of "running down the clock" to force legislators to choose between May's deal and a "no-deal" Brexit on March 29.
'You can't change it'
Guy Verhofstadt, who heads the European Parliament's six-member Brexit steering group, said the backstop clause was "absolutely needed" and there was hardly room to change the deal.
Going further, group member Philippe Lamberts was scathing.
"Saying you're against the backstop is like saying you're against bad weather. You might not like it, but you can't change it," he said.
Al Jazeera's Nadim Baba, reporting from London, said Tuesday's votes in the UK parliament did little to prevent a potential no-deal Brexit.
"MPs have said 'no' to a no-deal, without any clarity or consensus on how to stop it, just a green light to May to once again try to tweak that deal she reached with Brussels," he said.
"There have been dire warnings recently from businesses around the UK, warning of shortages of medicines as well as food. The EU is admitting that it is of concern to itself as well."
"But the message is, 'some things are more important than the economic hit we'll take' and that's code for, 'we don't want to do anything that will endanger the Good Friday Agreement, bringing back a hard border on the island of Ireland'."
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