A report by the committee examining whether it would be possible to avoid a hard border in Northern Ireland when the United Kingdom leaves the EU Single Market and the Customs Union found that little progress has been made to find a solution.
This year the foreign secretary, Boris Johnson, repeated this suggestion, saying the crossing the border would be as frictionless as travelling between two London boroughs.
The committee is made up of MPs from the Conservatives, Labour and the DUP.
In its report, the Northern Ireland Affairs Committee expresses concern over progress in finding a solution to the UK-Ireland land border post Brexit and points out the absence of a technical solution to render the border invisible. "Furthermore, we have no detail on how checks on goods and people will be undertaken away from the border", he said.
"During the implementation period, we recommend that the government works closely with its counterparts in Ireland and the European Union to develop an innovative border system capable of delivering customs compliance without traditional physical infrastructure at the border".
"However, we have heard no evidence to suggest that there is now a technical solution that would avoid infrastructure at the border".
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A policy statement from the committee directed to the United Kingdom and Irish Governments outlines six requirements for the final EU Withdrawal Agreement to meet the obligations of the Good Friday Agreement. Without agreement "then we are heading closer to the United Kingdom crashing out", one diplomat said. "They are desperate for their implementation period", one diplomat said.
"We have had no visibility of any technical solutions, anywhere in the world, beyond the aspirational, that would remove the need for physical infrastructure at the border", the report reads.
The UK's withdrawal from the European Union threatens equality of rights on the island of Ireland protected under the Good Friday Agreement, it has been warned.
The EU published its draft withdrawal agreement on Thursday with barely any changes to the protocol on Northern Ireland.
Last month Mrs May told British MPs that the EU's draft text of a withdrawal treaty was something "no United Kingdom prime minister could ever agree to" because it would effectively create a new border in the Irish Sea between Britain and Northern Ireland. The pro-EU pressure group Open Britain said the report should "mark the end of ministerial flights of fancy" over the Irish border.
A joint committee established under the peace accord to consider human rights issues on the island has insisted that the Withdrawal Agreement provides for the continuing North-South equivalence of rights, post-Brexit, as established under the 1998 Agreement.