An EU proposal for the Northern Ireland border threatens the "constitutional integrity" of the United Kingdom, Theresa May has said.
The EU's draft legal agreement proposes a "common regulatory area" after Brexit on the island of Ireland - in effect keeping Northern Ireland in a customs union - if no other solution is found.
Mrs May said "no UK prime minister could ever agree" to this.
The EU says the "backstop" option is not intended to "provoke" the UK.
Unveiling the draft agreement on what has proved to be a busy day for Brexit news, EU chief negotiator Michel Barnier called on the UK to come up with alternatives.
He said the text was "no surprise" and was just a legally-worded assessment of what had been agreed in the negotiations so far.
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What the EU's draft document says
The EU's 120-page paper covers the issues which have been negotiated so far - including the UK's financial settlement, citizens' rights and the future of the border between Northern Ireland and the Republic of Ireland - which is a member of the EU - after Brexit.
On how to avoid border checks in Northern Ireland, the EU says a "backstop" is needed in case neither of the UK's preferred solutions can be found.
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