The popular image of Brighton is a city where it is sometimes easier to buy a vinyl copy of a Funkadelic album than a pint of milk, and people really do while away the days in vegan coffee shops, dressed in corduroy loon pants. The reality, of course, is more complicated, but the idea persists of a retro-fixated would-be utopia that has precious little to teach the rest of the country. And yet, for the past two years, Brighton has been blazing a trail away from Westminster's tired three-party system towards a much more interesting way of doing politics.

At the 2010 general election the constituency of Brighton Pavilion elected Britain's only Green party MP, Caroline Lucas. Since 2011, Brighton and Hove city council has been led by the same party. The Greens have 21 councillors to the Tories' 18 and Labour's 14 (just to prove that Brighton is cooler than a lot of other places, the Lib Dems have none at all). For the past two-and-a-half years, they have formed a minority administration, pledged to making the city fairer and more sustainable, despite cruel cuts handed down by the coalition. With a slight air of overexcitement, their leading lights say they want to make Brighton like Barcelona, Sydney or Toronto, while other Greens seem to think it can be some sort of revolutionary hotbed a la Petrograd, or Turin – or, if necessary, Liverpool circa 1985.

And there have been unarguable successes. Thanks to the council's crusade for better pay, 102 of the 400 private UK businesses that pay the living wage are located here. The local Greens have embraced imaginative policies on housing and regeneration, and officially declared hundreds of acres of council land "open-owned". Brighton has become the world's first "One Planet" city, and much is made of plans for everyone there to "live well within a fairer share of the earth's resources".

What has played out here may well be a portent of life beyond traditional party politics, where a fiercely engaged, socially networked electorate send things spinning away from the usual way of doing things. The trouble is, it has been accompanied by an unholy series of battles that have set the Greens not just against their usual political enemies – witness an ongoing battle over raised parking charges and lowered speed limits – but each other. And now Brighton's radical Green future is in real danger of being snuffed out before it has even got started.

The Greens still affect an air of determined optimism. "William Gibson once said: 'The future's already here but it's not evenly distributed,'" says the Green's council leader (or, to use their preferred title, "convenor"). "Maybe we're seeing the future of politics in Brighton before the rest of the country catches up." His name? Jason Kitcat. The surname, he tells me, comes from the west country, and an era long before chocolate snacks and cat food. It meant that he was teased at school. "Heavily, yeah," he says, with a guffaw. "But it gave me the thick skin to be a politician now."

We meet just behind Brighton station, in the New England quarter, at a huge block of reconditioned shipping containers that a local housing trust has done up as flats. They will be used by people currently in homeless hostels: an example, I'm told, of "the Brighton spirit", which exemplifies creativity, ambition, and the idea that you can get things done quicker than some people think.

Jason Kitcat, Brighton Councillor … has had 'a few ups and downs'. Photograph: The Green Party

Kitcat is 35. He's a warm, occasionally self-deprecating presence, with a goatee beard that probably makes stereotyping him easier than it should be. His Polish-born wife, Ania, is also a Green councillor. He came to Brighton in 2000 to start an internet business, joined the Greens because he liked the idea of "thinking very long-term", and became the figurehead of the council in May last year. How has his first 18 months at the top actually been? "All councils are having a tough time," he says. "But I'm not going to deny that we've had some ups and downs, and as the only Green council, and the first Green council, there's been particular attention. And we have unfortunately provided some ill-advised stories which have not been helpful."

It is something of an understatement. This summer, as hundreds of thousands of people swarmed to Brighton to enjoy the tropical temperatures, the Greens hit big trouble. First, the time came for Kitcat's annual re-adoption as council leader, and one of his own colleagues sent a private tweet to the leader of the council's Labour group, in effect asking if he would support an anti-Kitcat coup. Inevitably, Labour was delighted, and it gleefully went public. ("It was a tricky time," says Kitcat. "But she apologised profusely, and I accepted her apology and we moved on.")

Another Green councillor took to his blog and gave the following glowing summary of Kitcat's record: "Jason Kitcat's policies have time and again betrayed working people, city residents – and the electoral interests of the Green Party of England and Wales."

That must have been nice. "I would have preferred it not to have happened," says Kitcat.

Then came the infamous bin strike. Kitcat's administration had resolved to belatedly bring the council into line with equal-pay legislation, but the process was handed over to council officers, rather than being worked through by politicians. As some saw it, he was washing his hands of responsibility for the proposed outcome – pay cuts of as much as £4,000 for some of the council's rubbish-collecting and street-cleaning workers. The result was a week-long strike, and strange scenes indeed: rubbish piling up in the streets and businesses being warned against clearing the streets themselves by a mysterious outfit called "the Brighton and Hove Talibin". Lucas supported the strikers; Kitcat was apparently nowhere to be seen.

Author and local resident Lynne Truss delivered a picture of what happened. "The place turned into Armageddon," she wrote. "Helped by foxes and the seagulls ... a tide of used teabags, eggshells, soiled kitchen paper, banana skins, smelly tin cans, and used sanitary towels (yes!) advanced in such a determined and menacing manner down nice residential streets, you could almost hear it breathing."

"It wasn't pleasant," says Kitcat. "It was very difficult. Of course it was. We didn't want to be there."

Caroline Lucas celebrates after becoming Britain's first Green Party MP. Photograph: PA Archive/Press Association Ima

And how did he feel about Lucas being on one side and him on the other? "Ultimately, we both wanted a fair outcome. The difference was, I was in the driving seat. From a negotiating point of view, I couldn't be out on the picket lines, could I?" He insists that contrary to allegations that he hid himself away, he "was in conversation on a weekly basis with the union officials, and all the officers involved".

He claims the eventual agreement as a success – but those who wanted to portray the Greens as a divided shambles had no end of ammunition, and they were helped by the party's noble refusal to adopt an orthodox disciplinary system whereby any rebels could be whipped into line. Some of them, it was claimed, were "watermelons": green on the outside, but red in the middle, and therefore dangerously leftwing. Others, such as Kitcat, were said to be "mangoes", with orange metaphorical pulp, denoting the cautious and centrist instincts of the Lib Dems. There was even talk of "figs" – green-and-black types – who were essentially eco-anarchists.

"There genuinely isn't a kind of hard split between different people, and fruits. It's not the case," Kitcat insists.

He laughs like a drain. "The media and the public seem to want authentic, free-speaking politicians, who seem genuine and honest, and not constrained in how they put forward their ideas. On the other hand, every time someone seems to break with the leaders or the party line, they're a rebel, and it's 'splits'. We're in a bit of a bind, aren't we?"

Next year, the Greens come to a big juncture. They will have to make around £25m of cuts from the council's budget, and the scope for efficiency savings will have run out. Kitcat says his party will "do the best we can for the people of the city", but the decisions required will inevitably be unpleasant. Some people think that pushing through such punishing measures will kill the Greens' lefty credentials, and it would be much better if they picked a watershed fight with the government.

North Laine district … the classic image of Brighton. Photograph: Alamy

Ben Duncan is one of the alleged watermelons, and the councillor who accused Kitcat of betrayal in his blogpost. A former journalist who now works for a Green party MEP, Duncan has floated a handful of provocative ideas, including a "tourist tax" on some of the city's bigger hotels, a possible boycott of one of the taxi firms opposed to the Greens' 20mph speed limit, and the possibility of Brighton allowing the opening of cannabis cafes and becoming the British version of Amsterdam. When asked on Twitter if he himself inhaled, he said this: "I only smoke weed when I'm murdering, raping and looting!" It was, he says, a reference to the famous anti-cannabis film Reefer Madness, but his political enemies didn't seem to get the joke.

Now, he is really on the warpath. There is mileage, he reckons, in the idea of the Greens following the lead of Trotskyite Labour councillors in 1980s Liverpool, refusing to set a cuts-based budget, and thereby putting Brighton in the vanguard of UK-wide anti-austerity resistance.

Appropriately enough, we meet in a city-centre cafe called the Red Roaster. "I think we need to do something that makes people think: 'Hang on – if you vote for the Greens you get something different,'" he tells me. "On behalf of people around the country, we need to make it Brighton and Hove versus this government, about austerity. We should either refuse to set a budget or try and set a budget that under the current legislation would be unlawful, and say to the government, 'Well, you come in and do it – you see how local people benefit from you sending a hit squad in to take over from elected politicians who won't deliver your austerity measures.'"

Kitcat isn't impressed. "It's absurd, frankly," he says. "Gestures like that will actually change nothing for residents, because the government will impose a budget on the council anyway." Duncan, however, says he will keep on keeping on – so one presumes that in the buildup to a final decision, it'll be mangoes and watermelons all round, again.

The leader of Brighton and Hove Council's Labour group is 46-year-old Warren Morgan, who looks like Al McWhiggin, the villain in Toy Story 2 (thick-set, glasses, neat beard). In September, he called on the Greens to "get a grip, or quit", said they were making Brighton "a laughing stock", and proposed a moratorium on all new Green schemes, "unless it can be proved that they have public support".

He has made a habit of going after the Greens on issues such as cannabis cafes and ill-advised tweets, which may indicate that by Brighton standards he's a bit of a square. At this suggestion, he emits a noise halfway between a groan and a smirk. "Speaking personally, I'm not the most hipster of people," he says. "Some people in Brighton would not identify with that image. It's a very, very varied city. It's not all hipster-cafe arts culture. And we try and represent everybody.

"The Greens' focus has been far too much on the city centre," he says. "They are not a party of the estates. They're a party of the city centre and the universities. There's nothing wrong with that electorate, but they need a broader view."

With Labour on 38%, the Tories on 25%, and the Greens on 21%, Morgan seems to fancy his chances of being the council's next leader. He and his comrades also want to use the Greens' travails to bring down Brighton's brightest political star, Caroline Lucas. If her seat fell to Labour, politics-as-usual would return to this patch of the south coast, and the short era of watermelons, mangoes and figs would be over.

"I have a lot of confidence in the people of Brighton," she tells me. "And I'm confident that they'll make a distinction where necessary … to be able to judge me on my record. I don't think you can hold an MP answerable for decisions they had no control over. People like to have an independent Green voice in parliament, and irrespective of their views about the local council, I think many people would be very sorry to lose that voice, sticking up for Brighton."

That might suggest she's trying to distance herself from her own party's local record, but when prompted, she comes out with a long list of Kitcat and co's achievements. She also talks about her own work in parliament: reminding Labour of its supposed principles and opposing the renewal of Trident, putting the case for the public ownership of the railways, and more.

But what happens if all that gets sacrificed on the altar of ill-advised tweets and the bin strike? It'd be a crying shame?

"It would be a crying shame," she says, and emits a mirthless laugh.

Does she think of herself as a mango or a watermelon?

"I've always refused to answer that question. I don't think it's a helpful question, and I'm not going to define myself as a piece of fruit," she says.

I've always liked to think of her as having watermelon-ish leanings.

"Thank you. But I'm not commenting on my fruitiness, to you or anyone else."