The shadow chancellor, John McDonnell, has warned colleagues if they want to avoid a "slow motion car crash" they have to change policy on Brexit.
So far, Labour's shadow cabinet hasn't shifted its position on Brexit despite devoting two meetings to the topic.
Labour wants the public to have a say via a general election - or if that's not possible, then the leadership says the option of a public vote to avoid no deal or a "hard Tory Brexit" is on the table.
In effect, that means a vote on any deal that is negotiated.
There had been a widespread expectation that the party would shift position towards offering not just a referendum on any deal - but that it would make explicit that Remain would be an option on the ballot paper and officially endorse Remain, while allowing MPs to dissent.
But this shift didn't happen on Tuesday.
A road to Remain?
Following disastrous European election results - overshadowed only by the Conservatives' poorer performance - deputy leader Tom Watson called for Labour to become an avowedly Remain party.
Significantly, he now has support from the shadow chancellor John McDonnell - who is usually closer to Jeremy Corbyn.
Mr McDonnell told me on Monday he felt it was time for the issue to be put back to the people, and he would campaign for Remain if there were a further referendum.
He also cited Harold Wilson's 1970s government which called a referendum, officially backed remaining in the EU, but allowed any members, and indeed ministers, who wanted to leave to call for withdrawal.
There's been pressure too for the leadership to adopt this position from others on the Left - notably the group Another Europe Is Possible, which is supported by shadow minister Clive Lewis and Momentum's national organiser, Laura Parker.
They say the time for equivocation and hesitation is over.
But apparently it isn't.
Hesitation, repetition but no deviation
At Tuesday's fractious shadow cabinet meeting I'm told Tom Watson and shadow foreign secretary Emily Thornberry pushed for "decisive action" - the latter saying "this is about leadership" - and John McDonnell was certainly under the impression that a decision was supposed to have been taken today.
Another left-winger, Diane Abbott, talked about rising discontent amongst the membership.
But in effect the decision to - at least - delay a policy shift was taken yesterday when unions affiliated to Labour met Jeremy Corbyn but came to no definitive conclusion on Brexit policy.
Both Unite and the CWU don't want to adopt a "Referendum and Remain" stance, partly for fear of alienating Labour Leave voters.
Some in the shadow cabinet agree - and party chairman Ian Lavery and shadow education secretary Angela Rayner expressed concerns about shifting position today.
So there will now be further consultation: first this week with members of Labour's ruling national executive - then with the unions.
Their general secretaries will meet again on 8 July.
'Muddle and confusion'
I am told that John McDonnell believes it's vital that a decision isn't delayed beyond this date as he and others want any Brexit policy to be clear well before a new prime minister is installed on 24 July.
Phil Wilson - one of the Labour MPs behind a failed Commons amendment on a so-called People's Vote - said: "Labour members and Labour MPs expect our party to have a clear policy that reflects our values.
"Instead, we have to listen to muddle, confusion and the sound of the can being kicked listlessly down a never-ending road."
His colleague Neil Coyle, who backs another public vote, had a union general secretary in his sights. He suggested the Labour leadership wouldn't shift position unless Len McCluskey of Unite would shift.
But the 26 Labour MPs - mostly, though not exclusively, from Leave areas - who signed a letter to Jeremy Corbyn last week warning against a Remain position and a "toxic further referendum - will be pleased that pressure to move quickly has been resisted.