WOLVERHAMPTON (Reuters) - The prime minister’s battle to keep his country in the European Union - and save his political future - may depend on the campaigning power of his fiercest rivals: the Labour Party.

Britain's opposition Labour Party leader Jeremy Corbyn speaks on the BBC's Andrew Marr Show in London, January 17, 2016. REUTERS/Jeff Overs/BBC/Handout via Reuters

On June 23 Britain will vote on whether to withdraw from the EU in a referendum that has scrambled the traditional left-right battle lines of British politics.

Cameron has staked his political legacy on being able to persuade Britons to stay inside the 28-member bloc. But his centre-right Conservative Party is divided on the issue, forcing it to mothball its campaign machine and adopt a neutral stance.

The leftwing Labour Party has overcome the eurosceptic tendency of party leader Jeremy Corbyn - who voted to leave the EU’s predecessor in a 1975 referendum - and agreed to throw its weight behind a pro-EU campaign.

With public opinion finely balanced and up to one in four voters yet to make up their minds, it could be down to Labour to deliver Cameron, who will step down before the next parliamentary election in 2020, a referendum victory.

Labour has well-established influence across Britain’s cities and an experienced campaign machine that specialises in targeting winnable votes and making sure supporters cast their ballots on polling day.

“We’ve got the foot soldiers,” said Labour lawmaker Emma Reynolds. “I see our job, and our responsibility as a Labour Party, in this referendum to talk to Labour voters ... to try to persuade them to vote ‘In’ and then get them out to vote.”

Anthony Wells, director of political polling at YouGov, said that could be crucial. “Leaflets and literature are the sort of things that change minds, and knocking on doors and talking to people is all about energising people - getting the vote out and people to commit,” he said.

“It’s not sexy, and it’s the boring administrative stuff behind any election, but it matters.”

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Labour lost a general election in May to Cameron’s party, but still won more than 9 million votes - 30 percent of all those cast.

Reynolds is one of three Labour lawmakers representing Wolverhampton, an industrial city in central England that is typical of many Labour strongholds outside the more prosperous and largely Conservative south.

Labour has a majority of seats at local government level in England’s five biggest cities and the party represents more than two thirds of inner-city parliamentary constituencies across London, Birmingham, Manchester, Liverpool and Leeds.

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The national Conservative Party decided last year to take an officially neutral stance, faced with the risk that a centrally-run campaign could drive a deep wedge between its pro-EU and eurosceptic factions.

As a result the party’s financial muscle, paid staff, regional offices and treasure trove of voter data - key elements of its unexpectedly successful May general election campaign - are all off limits when it comes to the EU vote.

In Wolverhampton, the local Conservative organisation said it had not discussed or agreed upon a unified position for the referendum.

“We are not forming any general view as it is up to individuals to decide for themselves,” said Conservative councillor Wendy Thompson in an emailed response to an interview request.

Labour’s campaign is expected to be a stark contrast.

A team of around 70 activists knocked on over 10,000 doors for Reynolds’ cause in a 2015 general election, using a sophisticated voter identification system to target the households most likely to vote Labour. On polling day volunteers hit the phones, calling constituents to remind them to vote.

The same strategy will be used for the EU referendum, leveraging the party’s political influence to deliver a message highlighting the EU’s impact on job security and workers’ rights.

On Saturday, Reynolds and her team will be manning a street stall in Wolverhampton armed with Labour-branded literature produced by the party’s headquarters - one event in what will be a national push to promote the centrally coordinated ‘Labour In For Britain’ campaign.

“I like where Labour is going, so that would sway me,” says Ruth Ashton, interviewed by Reuters outside the Wolverhampton university where she works. She is likely to vote to stay in the EU and says she expects her pro-Labour friends to do the same.