Nigel Dodds, the Democratic Unionist Party's Westminster leader, said on Tuesday, December 5, the government had now reassured them the constitutional integrity of the UK will not be undermined.
Mr. Dodds said the DUP only received a written text of the deal which had been proposed after discussions with the Irish government late on Monday morning, December 4.
The DUP's leader, Arlene Foster, said it was unacceptable.
Labour's Shadow Brexit Secretary Sir Keir Starmer said that was the moment when "fantasy met brutal reality."
"The DUP tail is wagging the Tory dog," he said in Parliament on Tuesday.
Call for Halt to March 2019 Brexit Date
Mr. Starmer also called for the government to drop its plan to cement the Brexit date of March 29, 2019 in law.
It has emerged that Mrs. May spent 20 minutes on the phone to Ms. Foster on Monday trying to assure her the deal did not constitute a threat to Northern Ireland's position within the UK.
Britain's future must not rest with this pair. We are far far better than that. pic.twitter.com/XtMjpUe51C— Keir Starmer (@Keir_Starmer) 2 December 2017
The DUP traditionally represents a hardline Protestant and loyalist electorate in Northern Ireland which has long resisted a united Ireland.
But Mr. Dodds accused the government of the Republic of Ireland of taking an "aggressive and anti-unionist" stance.
He said the "regulatory alignment" suggested by the Irish government was not necessary and the DUP preferred other options, such as trusted trader schemes.
Mr. Dodds said the text of the agreement must "translate the principles of what has been negotiated".
Mr. Dodds denied the DUP had a veto over the Brexit talks and said it was Dublin which was acting in a "reckless and dangerous way".
He claimed Irish government policy had taken a more hardline stance since Leo Varadkar took over as Taoiseach, or Prime Minister, from Enda Kenny.
Asked if the DUP would tear up the confidence and supply agreement which allows the Tories to rule day to day without the need for fresh elections, Mr. Dodds dodged the question but said he was confident Theresa May would not do anything which would separate Northern Ireland from the rest of the UK.
He said he did not want the UK government to walk away from the talks but wanted the government to honor its "red lines" which, he claimed, were the same as the DUP's.
Brexit Secretary David Davis defended the regulatory alignment idea.
'Alignment Isn't Harmonization'
"The presumption of the discussion was that everything we talked about applied to the whole United Kingdom. I re-iterate: alignment isn't harmonization, it isn't having exactly the same rules. It is sometimes having mutually recognised rules, mutually recognised inspection, all of that sort of thing as well. And that is what we are aiming for," Mr. Davis told Parliament.
Mrs. May was supposed to reach a vital agreement on Monday with the European leaders that "sufficient progress" had been made in Brexit talks, paving the way for the EU summit later this month.
The summit, which is due to begin on December 14, would have become a crucial step in the ongoing Brexit negotiations, as it was supposed to allow Mrs. May to begin long-sought trade talks with the EU.