Rejecting what he saw as "naked discrimination of a public sector position" and a "statement of intent" to introduce a hard border, Sinn Féin's Newry-Armagh MLA Conor Murphy said: "With this British Border Force recruitment drive, the British government are clearly planning to impose their hard Brexit border across the island of Ireland".
But a version published by the European Union in February contained only the third "backstop" option, effectively drawing a customs border down the Irish Sea, which a furious Mrs May said "no British Prime Minister could ever agree".
EU diplomats said it was created to resolve the Irish border issue by avoiding customs checks between Britain and the EU, while keeping Britain out of the EU's customs union - something May has also rejected despite pressure from some in her own party.
While talks will continue over the next three weeks to try to achieve a breakthrough, there are fears that the Irish border question could, once again, prove a roadblock in wider Brexit negotiations when European Union leaders meet again in June.
Negotiations on Brexit are starting to reach a critical point, however, in reality for the people of Northern Ireland it's very simple, we are not prepared to go backwards, which means in effect no borders, north, south, east or west.
Its objection to the customs arrangement proposal apparently reflect a reluctance to turn a blind eye to possible irregularities along the Border.
The United Kingdom has committed to keeping Northern Ireland in full regulatory alignment with the European Union in order to avoid a border with Ireland.
The Commission's proposal would effectively cut Northern Ireland off from mainland Britain and May has said no British prime minister could accept that.
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"Without a solution, there will be no withdrawal agreement and no transition".
Sinn Féin MLA Conor Murphy said the requirement reinforced the belief that the British government was "clearly planning to impose their hard Brexit border across the island of Ireland".
The comments from Donald Tusk couldn't be clearer - the European Union are not bluffing. "And we are continuing an intensive work programme to engage on all the scenarios set out in the joint report", the British government spokesman said.
But the customs union vote was also only the beginning of what will likely be a war of attrition on all aspects of Brexit between the Lords, where May does not have an overall majority, and the Commons, where the government's majority only exists because of the smaller DUP party of Northern Ireland.
Mrs May said in her speech last month that remaining in the customs union would "not be compatible with a meaningful independent trade policy".
Marina Donohoe, Enterprise Ireland's director of the United Kingdom and Northern Europe, added: "The Northern Powerhouse has proven its powerful growth potential, now growing faster than London".
The Telegraph reported that two proposals were given.