Caroline Lucas is sitting on the curve of a trendy sofa inside Liverpool's ACC Conference Centre. Even close up, she has to speak loudly so her words can be heard above bustle of delegates - from France, South Korea, even Mongolia – who have gathered at this meeting of Global Greens. Their concerns range from ocean conservation to corruption. But the challenge on Caroline’s mind is closer to home.

On the Saturday afternoon, the Green Party of England and Wales passed a motion in support of electoral alliances for proportional representation. The aim is to work together with other parties to beat the Tories and secure the introduction of electoral reform. If enough reform-minded MPs are returned at the next general election, legislation could then be passed and a second election held shortly afterwards under the new system. "That's the prize," Lucas says.

So how exactly could it work? "We're really clear that this is a bottom up, locally-led process," says Lucas. "It could be about standing down, it could be about having an open primary." Or simply targeting resources diferently in different seats. She later adds in writing: "At the heart of any agreements will be an understanding that each party has its own distinct offer to the electorate, and that any one off pacts have electoral reform at its core."

The 2016 by-election in Richmond, triggered by the decision to back a third runway at Heathrow, has become one example of how such alliances can benefit the Greens – even in cases where they are the ones stepping aside. According to Andree Frieze, the potential candidate in Richmond, her decision to stand down protected the Green party’s “integrity”. It also won an assurance from the revived Liberal Democrats, who went on to win the seat, that they would talk seriously about sharing wards in 2018.

The challenges, however, are big. Firstly there is the problem of keeping individual party identities intact. Even the wording of the Green's new motion took some care: “The language of the motion was very specifically about electoral alliances, not progressive alliances, because we could spend hours and hours defining what we mean by progressive and that probably isn’t terribly fruitful,” Lucas explains.

There are also now a number of different organisations with similar intentions, from local groups like Sussex Progressives, to larger organisations like Makes Votes Matter. A new Progressive Alliances Network, known as Pan, is trying to steer these bodies more closely together, but it is still early days.

Questions also remain over how far promises will be kept (on both electoral alliances and on delivering proportional representation). There is no certainty, for example, over when the Lib Dems will return the favour the Greens paid them in Richmond. It is certainly unlikely to be the up-coming by-election Manchester Gorton – where they hope to reclaim the second place that the Greens won in 2015, or even take the seat itself from Labour.

Yet Caroline is not disheartened. “Having come second [in Manchester Gorton] before, that’s a really key opportunity for us to go really strongly," she says. "Some of the quid pro quos are not going to be direct in time or location necessarily. It could be that later down the line, in another area, that [the Lib Dems] will do something that’s positive in return”.

She also has stern words for those don’t keep their pledges – especially when it comes to electoral reform. Of Justin Trudeau, the Candian Prime Miniser, who recently backed out of a commitment to reform his country's electoral system, she is wary.

"I think what Justin Trudeau did was absolutely outrageous and something that should give us all good reason to be very cautious," she says. In some countries, she notes, reform-minded parties demand written agreements called memorandums of understanding. "But I also think that in some countries that have worked together they’ve had written Memoranda Of Understanding. They’ve been very public about those MOUs, so it makes it harder for other parties to go back on promises that they’ve made.”

The bigger problem is arguably persuading Labour of the merits of proportional representation. The Greens would gain much from the switch; their current 1.1m share of the 2015 vote is vastly under-represented by their sole Westminster MP. Whereas for Labour there has traditionally been a strong desire to protect its status as a majority party.

But signs of change are on the rise. My colleague Anoosh has reported on the new generation of Labour members who are keen to change the structure of British politics. A number of senior Labour and opposition politicians have joined the Greens in their call for a cross-party alliance. News of the electoral alliance motion’s success has been welcomed by the MP Clive Lewis, who tells me: “I welcome the Green Party vote to form alliances against our common enemy - the Tories.”

Across the progressive spectrum, hope is building that similar, constituency-by-constituency alliances could bring down yet more Tory MPs. Speaking on a panel earlier in the day, Neal Lawson, director of Compass, compared the struggle to Star Wars: “Let me announce here this morning, on 1 April, that it will be called ‘The Rebel Alliance’ and George Lucas the director of the Star Wars films has been in touch and has already cast for the remake of the film […] and Caroline, of course, is Leia.”

Ultimately, Lucas believes, success will come down to trust. "We can either go forward thinking the worst of people, and thinking that this is not worth pursuing because people might not keep their word, in which case we are no further forward than we are today," she says. "Or we can try and reach for ways of having a new kind of politics, and it does genuinely feel as if there’s more interest in this than ever before.”

She is loathe to say that the “extreme” Brexit which Theresa May appears to be pursuing is good for anybody; any gains opposition parties might make come at too high a price. But as silver linings go, it may also be giving collaborative thinking greater force.

In the Netherlands, the Groenlinks party soared to become the biggest left-wing party in the latest elections. Their digital strategist Sybren Kooistra thinks this is partly due to a Brexit backlash: “Brexit helped us because it sparked the feeling that we need to do something now.”

There is also much talk at the conference of deepening new forms of alliance with Europe. “There is a very strange reality: it is that we are adapting to the fact that the UK will soon be no more in the EU,” says co-chair of the European Greens, Monica Frassoni. “But this is not at all the situation with our Green people, with the NGOs, with the civil society organisations, with the peace movements […] That is why I am of the opinion that in reality we will not lose them. And that is why we also believe that this thing of Brexit is somehow not final. It’s not the end of the story.”

Intenational conferences like these make clear the many challenges of co-operation across borders, from delegates who don’t speak English, to those who were denied visas. But the Greens seem committed to pursuing alliances across divides of all kinds. In this way - as in Star Wars - there are signs of A New Hope.