Labour, Liberal Democrats and Conservatives have all had a go at running Bristol City Council.

They even tried running it together a few years ago. That “rainbow coalition” worked ... for a while.

On Tuesday night a unique drama unfolded, as yet another minority administration went down in flames.

What made it exceptional was that this happened during the annual budget debate.

And it was a budget amendment put forward by the Lib Dems – one of a number they proposed – that felled their rivals.

The subject was waste treatment, a few hundred thousand pounds in Labour’s proposed £365-million budget to take long-term plans further.

Harmless, you might think. The problem is that these plans could – I stress, could (not would) – have included the possibility of a mass-burn incinerator at Avonmouth.

The irony about these plans is that they are the work not of Bristol City Council alone, but of its two Tory neighbours, North Somerset Council and South Gloucestershire Council, too.

Why ironic? Because it was the small Conservative group on Bristol City Council which put a spanner in the works by voting with the Lib Dems to knock that waste spending out of the budget.

Why did they do it? They think they are in with a good chance of winning the Bristol North West Parliamentary seat at the next general election.

Bristol North West includes Avonmouth. The Conservatives do not want to be seen to be attached in any way to plans that could – I stress, could – one day see an incinerator built there.

But how does all this lead to Labour resigning power at Tuesday night’s council meeting?

First, Labour genuinely believes in the current buzzword, “partnership”.

Never mind that the neighbouring councils are Tory-controlled. They regularly work with Bristol – and with Bath and North-East Somerset – on a variety of issues.

It’s called the West of England Partnership and it brings the local authorities together to tackle problems in transport and waste treatment that they feel they can’t handle alone.

Labour felt it would be wrong to pull out of a strategy agreed with its partners – not least because the penalties for doing so are unlikely to be cheap.

But Labour leader Councillor Helen Holland and her finance supremo, Councillor John Bees, also genuinely believe in the partnership’s policy of seeking Government cash, in the form of Public Finance Initiative (PFI) credits, to foot half the bill for one or more major treatment plants.

Even so, Labour could have ridden the storm – ignored the opposition, if necessary – if only it was not a hung council.

But it is. Those old days of bumper Labour majorities year in, year out have long gone. Bristol City Council is chronically hung. No party has a majority – at least 36 out of the 70 seats.

When Labour took control two years ago, they had not exactly been swept to power at the May 2007 elections. They took two seats off the Lib Dems, but that still left them on 25, six behind their rivals.

(The gap widened to 24 and 32 when the Lib Dems beat Labour at the St George West byelection last October).

But that slip at the polls was enough to see Councillor Barbara Janke toppled, not only as leader of the council (in favour of Labour’s Councillor Holland) but as leader of the Lib Dem group (in favour of Councillor Steve Comer).

Councillor Janke regained control of her party last year and, following Tuesday’s dramatic events, finds herself leader of Bristol City Council for a third time.

But for how long? Don’t forget, the Tories’ slightly maverick leader, Councillor Richard Eddy, has played the kingmaker before.

He pulled the rug from under Labour’s feet by voting with Councillor Janke. But on another issue things could go the other way.

The Liberal Democrats know that so long as they are in a minority, Councillor Eddy could do it again, unseating them this time.

What they have to do is make real gains in the local elections in June. That will be the focus of their next three months in power.

But the electoral arithmetic is far from straightforward. It’s not the whole council that is up for grabs. It is just 23 seats – roughly a third – in 23 of the council’s two-member wards.

No fewer than 11 of these are held by Lib Dems now. The party has to hold all of them and gain at least four others.

It’s a ludicrous system that gives us a critical election in one part of Bristol and a holiday from politics in the very next street.

But that is what happens. We elect just a third of the council every year with a “fallow season” every fourth year (like 2008), when there is no poll anywhere in the city.

It makes it harder to gain a majority because the winds of change blow in different directions each year. Labour’s in favour in one year, let’s say, so perhaps it gains a seat or two in the south of the city. Another year, with Labour deep in the mire, we’re now looking at another part of Bristol and fewer of the party’s seats are vulnerable.

The “thirds” system of voting also means no party really has a mandate – and no one in charge can get on with the job without having one eye on the next round of elections.