Kirklees Council currently has no leader after Labour’s nominee failed to muster the backing of the other parties.
Councillor Shabir Pandor was in the running for the leadership after fellow Labour members picked him for the top role over ousted David Sheard.
But Coun Pandor lost out by 33 votes to 31 after the Tories, Lib Dems and Greens all refused to back him.
Green and Liberal Democrat leaders Coun Andrew Cooper and Coun Nicola Turner raised trust as an important factor in their parties’ decisions.
Four Labour councillors, including Coun Sheard, were absent at the annual council meeting so did not vote.
Each party leader conducted a 15 minute meeting with Kirklees Council’s chief executive Adrian Lythgo following the vote to broker a deal.
But no compromise was made – meaning that Mr Lythgo will run the council until a plan is formed over the coming weeks.
Mr Lythgo said: “Group leaders have agreed that tomorrow they will consult their own groups ona date to come back for an extra council meeting to vote on a new council leader.
“Between now and that meeting, conversations will take place within groups to agree a potential way forward.
“The constitutional position is that in the absence of an agreed council leader, I will run the council. I will make arrangements to consult group leaders on decisions that need to betaken in that period.”
After he was nominated by Coun Jean Calvert – who was sacked as deputy leader by Coun Sheard just this week – Coun Pandor appealed for those who planned to vote against him to change their minds.
But he was roundly criticised by each opposing party leader.
Kirklees Green Party leader Coun Andrew Cooper said: “Trust is vital in the role of a leader. Trust in their ability and trust in what they say.
“We understand that the Labour Group were told by their new leader some days ago that our support for his administration had been secured. It had not.
“If a leader cannot tell the truth to their own political colleagues, no matter how unpalatable that might be, then how can he be trusted with the leadership of this council?
“This bid to lead Kirklees Council is not built on principles, policies or a programme.
“It is a bid constructed of personal ambition, fuelled by promises and secured by patronage. It cannot stand. It cannot be supported.”
New Tory leader David Hall said a council leader “needs wisdom above cleverness, [to be] a solid negotiator, not a wheeler-dealer.”
He then confirmed that the Conservative group, which was missing one member, would not vote for Coun Pandor.
Lib Dem leader Coun Nicola Turner also told the council chamber in Huddersfield the Coun Pandor did not have the backing of her party.
Labour councillors David Sheard, Graham Turner, Cathy Scott and Judith Hughes did not turn out for the vote.
Coun Pandor was offered the chance to respond following the leaders’ speeches by the new Mayor, Coun Jim Dodds, but declined.