The Tories are producing fantasy "non-solutions" to the Irish border, Sinn Fein said.
They are indifferent to the fallout from Brexit for Ireland, party president Mary-Lou McDonald alleged.
The border between Northern Ireland and the Republic will be the UK's only land frontier with an EU state after next year's withdrawal and is one of the most vexed issues facing negotiators in Brussels.
Theresa May's Cabinet is considering whether to support a customs partnership with the EU after Brexit, whereby Britain would effectively collect tariffs on behalf of the bloc, or a technological solution to border control known as "maximum facilitation".
Ms McDonald said: "Allowing the Tories to dodge, equivocate and engage in fantasy non-solutions to the Irish question is most certainly not the answer.
"If the architects of Brexit still can't agree what Brexit looks like, that is their problem.
"It will not become Ireland's problem."
The British Government has said it is committed to ensuring trade and the passage of people and services across the island is "frictionless" after next year's withdrawal from the EU.
The intention is to avoid installing customs posts on the border.
Brexit Secretary David Davis has attempted to allay the fears of cross-border businesses concerned about the impact of customs checks at the currently free-flowing frontier.
He has suggested a technological solution or a trusted trader scheme whereby paperwork would be cleared in advance.
The Prime Minister has pledged no return to the heavily-militarised border of the past during the Northern Ireland conflict.
The expectation is that a deal would need to be struck between European leaders and Britain by this autumn to avoid the UK dropping out of the union without an accord.
Ms McDonald told her party's ard fheis (annual conference) in Belfast that this month's summit of European leaders in Brussels represented "crunch time".
The republican leader warned there could be no progress to the next phase in negotiations without an answer to the Irish question.
She said: "This challenge is as real for unionism as it is for republicans and nationalists.
"Treading water, fingers crossed and hoping for the best is not a strategy at this time.
"Propping up a Tory government that glories in wilful ignorance and indifference to Ireland is not the way forward."
The DUP's 10 pro-Brexit MPs are supporting Theresa May's minority Government in key votes.
Ms McDonald added: "If the British Government don't have a credible, acceptable proposal;
"If they cannot demonstrate how they will avoid a hard border, protect the Good Friday Agreement and citizens' rights, then the Irish Government must call them out.
"There can be no question of progressing to the next phase of these negotiations in the absence of an answer to the Irish question.
"This is one test that our Government cannot flunk."
Sinn Fein is pressing for Northern Ireland to remain part of the EU Customs Union after the exit, and points to the majority in favour of Remain from the country during the referendum vote.
If a Brexit deal is not done, a time-limited backstop option - which the EU interprets as Northern Ireland remaining in the Customs Union - would be introduced.