One pro-Brexit campaigner said the fear was the backstop plan would put Britain "in purgatory", essentially still in the EU's customs union indefinitely.
London said the backstop would only be a last resort and would end by December 2021. He added, "But a temporary backstop is not what we or, I believe, the Northern Irish people, want".
Barnier has proposed that Northern Ireland effectively stays in the single market and customs union following Brexit, a suggestion that has prompted claims that the EU is seeking to annex Northern Ireland.
Labour said that among the amendments was one that removed the requirement for the government to hold a "meaningful vote" in parliament on the deal it will try to negotiate with the European Union for Britain's future relationship with the bloc.
In essence, the Guardian explains, the UK's backstop or fallback proposal is for the whole of the United Kingdom to remain in the customs union for a limited period after the end of the transition period - so it would leave the EU in March 2019 and the single market in December 2020, but stay in the customs union for a while longer.
"The UK is clear that the future customs arrangement needs to deliver on the commitments made in relation to Northern Ireland".
Others questioned the substance of the proposal.
London says this is unacceptable, offering instead a "temporary customs arrangement" for the whole of Britain that would "maintain the status quo for traders in respect of customs processes". That means no border posts or customs checks after Brexit to protect economic links and the hard-won peace on the island of Ireland.
Best for Britain said a referendum could be held in early 2019 and the European Union would allow Britain to have a second referendum as late as February or March.
Both sides want a deal by October to avoid barriers on the Irish land border that might fuel new conflict in the north.
It would be used if a permanent customs deal is not in place at the end of the 21-month Brexit transition period, with the aim of avoiding a hard Irish border.
But Mr Barnier said the proposal raised a number of relevant and hard issues.
Barnier said that while some progress has been made, several issues remain to be resolved before the 28-29 June EU summit.
"We are today at a time of decisions and choices".
"Before we break for the summer in the Commons we'll be taking the other two bills that need to be addressed that are relevant in relation this issue", Mrs May said.
But he said: "Theresa May and her team have agreed to the backstop in the March agreement and there is no question of backtracking on that".
Just 0.85 percent responded that they had previously wanted Northern Ireland to join Ireland and now wanted to remain British.
The news comes alongside the reveal of Origin Access Premier, which will give access to even more games in the EA online library. That has been a constant complaint by gamers, in fact, and something Electronic Arts has made a decision to address today.
Doria Ragland was close to tears as she watched her daughter Meghan Markle walk down the aisle to marry Prince Harry in May. One guardsman stood out: Charanpreet Singh Lall , a 22-year-old Sikh from Leicester , who wore a black turban.
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