This happened in Lambeth in 1982. The Conservatives and Liberal/SDP Alliance tied with Labour - but there was a Tory Mayor, so the Tories/SDP/Lib Dems took control.
The two key wards here were Princes and Bishops in the Waterloo/Kennington area.
One of the nastier early events of the 1982-86 council was how Ted Knight's Labour group bullied Gordon Ley, the top voting SDP member elected in Princes Ward.
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Ley ran a shop in Lambeth Walk and was vulnerable to pressure. He had his shop front smashed in and other acts of intimidation - and being made an offer he couldn't refuse walked the floor, and joined the Labour group.
This restored Labour control, Ted Knight's glorious reign was only interrupted by a couple of months.
If you want to see what happened at the end of Ted Knight's chaotic period where he enforced extreme policies by his charisma and if necessary brute force with a fag-paper thin majority check this video.
At the point I've set it there is a councillor with his arms in the air - who it is rumoured either voted to set a rate. or abstained.
Curious thing is by this time the council had refused to set a rate in 1985, the District Auditor had estimated the cost of late rate-setting to be £125,947 and surcharged the Labour councillors this amount between them.
Lambeth and Liverpool followed similar paths here - and it split the party nationally between the Kinnochites and the Bennites.
The video here, which is from LWT "The London Programme" shows part of the 1986 rate-setting meeting, and in fact some sort of face-saving deal must have been done to avoid bankrupting the council at this point.
Looks like Mike Bright - who holding his hands up in shot - actually voted to set a 1986 rate, and we know another Labour councillor also did so.
Meanwhile Ted Knight and the 29 remaining Labour councillors refused to vote to set a rate, knowing that as they had lost their appeal against disqualification none of them could could continue to serve as councillors.
Meanwhile Linda Bellos had been chosen to succeed Ted Knight as soon as the 1986 May elections were held (assuming Labour won) and Labour would then have a new left wing council leader - and she would have a non-bankrupt council to run.
Puddy_Tat is right - sometimes not a stable way to run a council!
I asked a contemporary to review my comments above for accuracy, and there are a couple of amendments and some further remarks:
1. Hastings situation compared to Lambeth 1994
Hasting council seems to have reached the same point that Lambeth did in the 1994 hung council, where there was potentially a three way deadlock in which any two parties could defeat the third. That did not happen in Lambeth because there was undeclared coalition between Labour and Lib Dem, which provided a stable majority and allowed for rotating chairing, What seems to have happened in Hastings was that, with rotating chairing non longer available under the cabinet system, there had been a power sharing agreement including Labour and greens, who had 20 councillors between them out of 32. Labour, according to the greens, dropped the deal - but as a result of pressure from the national party - leaving the reform councillor as king maker.
Lambeth 1982 - 86 more information and difference nuance
Casting vote of the Mayor
regarding the Tory Mayor in 1982 and the casting vote. The first council meeting of the 1982 session, the Mayor making, was chaired, as was and probably is the rule, by the preceding outgoing mayor, and he was not a councillor in the new council. He was “Johnny” Johnson, not a Tory but Labour. Only a Labour member could have been Mayor at that time. He was a working class councillor, a postman I think, of what might be called the right wing labour stripe. He had some personal grudge against Knight. So he used his casting vote, to vote in a Tory Mayor Hugh Chambers, a councillor, who thereafter held the casting vote.
Mike Bright's reputation saved - how Lambeth retianed socialism under Linda Bellos plus Trotskyite tit-bits
Mike Bright did not vote for a legal rate. Two Labour councillors, Janet Boston and Stuart Cakebread broke ranks, both were barristers, and the budget was adopted 32 votes to 31. I see that Cakebread has just lost his own case suing a client for fees, under a share of the damages deal, in the sum of seven million.
Other Labour councils who had earlier refused set rates, had done so, prior to Lambeth. including Sheffield then lead by a firebrand left winger, Blunket, the man who made subsequently made it easier to lock people up and throw away the key.
Mike had resigned prior to the rate setting. if he had remained the vote would have tied and gone on the Mayors casting vote. The resignation caused a by-election, won by Linda Bellos. Nonetheless Mike was surcharged and barred: those penalties covered the earlier delays in setting the rate and the consequent interest cost to the council.
I have not looked at the video, but post June/July 1985 after the rate was set there was indeed a workaround. Prior to be being disbarred, they set up a special committee, composed of Ms Bellos and one or two other Labour councillors (Kingsley Smith) who had been elected at by-elections, and were not threatened with disqualification, and all the powers of the council were transferred to the special committee so that control could not pass to the Tories, in the time before the 1986 election. By the way, Christine Headley (a Liberal - Streatham Hill) was elected in a by-election in 1984. (says my correspondent)
Bellos selection in Larkhall, may well have been galling for Millie Haston . Millie and her husband Jock, were long-term Larkhall ward Labour Party members. They had a big house on Larkhall rise. I think their prosperity was down to Millie who had a fashion business.
The Hastons had been Trotskyite comrades of Gerry’ Healey, and pioneers of Trotskyism in Britain. She was one of the little group which included Ted Grant, the long term guru of Militant and also her husband prior to Jock, Ralph Lee, - Jock was gaoled during the war for organising a strike. The two famous members of the group were T.D Smith and Michael Tippet (although Tippet's Wikipedia entry only mentions the Communist party).
Millie Haston moved to the Labour right, and she was a bitter opponent of the Healyites.
There is a lot of material around about these Trotskyite people now, including a biography of Ted Grant. See also:
James Ritchie (Jock) Haston (1912-86): Trotskyist Leader and Workers’ Educator by John McIlroy
I don't think anyone will get more detailed than that!
Edited: correction to rate-setting vote figure.