What self-respecting lefty wants to line up alongside Boris Johnson, Nigel Farage and Iain Duncan Smith? Labour supporters may be divided over the issue of Europe, but if undecided voters swing to Brexit they could hand victory to the rightwing rabble that leads the charge for a “sovereign” Britain. And there’s the rub.

“My view of the EU has always been not that I am hostile to foreigners but I am in favour of democracy. I think they are building an empire and want us to be part of that empire, and I don’t want that.” These are not the words of John Redwood, but of the late Tony Benn.

There are good, principled, leftwing reasons for questioning our membership of the EU, that bloated, undemocratic bureaucracy. It is not Little Englander-ish to question whether Britain could not offer a better deal to its populace were it to depart from a parliament that makes even George Osborne look good.

But that’s just the tip of the iceberg. For many, the treatment of Greece and Syriza was the moment the EU crossed the Rubicon. A radical, leftwing government with a mandate to oppose austerity was crushed by the European machine. The latest Greek youth unemployment rate stands at 48%. Nearly half their young people are jobless. This is not Jacques Delors’ “social Europe” – a “socialist conspiracy” that enraged Margaret Thatcher and Rupert Murdoch. Today’s EU is a club enraptured by neoliberal economics with no answer for when that, inevitably, fails.

If that is not enough to shift the thinking of the leftish segment of the 10 million undecided Britons, then perhaps they should look closer to home, where the EU’s malign influence is often felt. Privatisation, for example, is broadly encouraged by the EU. Far from seeking to protect member states’ national assets, the EU demanded that nationalised postal services be opened up to competition, and finally removed any protected areas which could be subsidised. This forced the hand of government, which could no longer prop up the Royal Mail to compete with private competitors – ensuring its privatisation. It is no coincidence that a rash of publicly owned postal services had to be offloaded across the continent.

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And then there is TTIP, the secretive trade deal between the EU and the US, a major bugbear for those on the left. It allows corporations from across the Atlantic access to each other’s markets and, with the “investor-state dispute settlement”, threatens to give private corporations the ability to take governments to court if their profits are infringed upon. A big US medical firm isn’t given access to pitch for business within, or even compete alongside, the NHS? The government could be sued, and privatisation forced upon us. This seems unlikely – defenders have said this will never come to pass, although David Owen clearly thinks it is a possibility – but the EU’s reluctance to allow open scrutiny of the deal only feeds suspicion and mistrust.

But even so, the EU already prevents government from allowing British companies any special status when tendering for contracts. This has the patently absurd consequence that steelworks are closing across the land while we import foreign steel with which to build warships. But there is no option when fair and open competition puts the interest of business ahead of that of the nation.

Then there’s the issue that the left, and Labour particularly, has been so keen to ignore: immigration. And the unfettered access to British jobs that EU citizens enjoy. Of course it’s a two-way street, and European immigrants do much to boost the economy – almost certainly a net positive. But Farage and co have done great work targeting traditional Labour voters because there is a grain of truth to what they say. Britain does not have the ability to limit the number of people coming in. That is not to mention the inherent unfairness, from a leftwing perspective, of the current deal. If you believe in open borders, free trade, movement of workers, then why should a Polish builder have any more right to trade in London than one from Djibouti?

Cameron, Osborne and possibly Johnson would have free rein to roll back the important protections EU membership offers

In any case, an influx of eager workers from poorer countries inevitably puts deflationary pressure on British wages. And business will take advantage. The only counter is an eye-watering minimum wage – far beyond Osborne’s “living wage” – coupled with regular inflationary increases. Yet, not only would the business community be in uproar – making it politically impossible – it would only further increase the draw to Britain. In any case, it is unlikely to be something a Tory government would even consider.

And given who controls the country currently, how can any leftwinger, with good conscience, believe that Britain will be any better off outside the EU when it will be Cameron, Osborne and quite possibly Johnson who lead negotiations on what our relationship with Europe will look like thereafter? They would have free rein to roll back the many important protections that EU membership offers us.

Workers’ rights, human rights, police coordination, inward investment in the arts, science, sport – all important EU benefits that there’s little evidence the Tories value and would seek to preserve. If you judge a person by their friends, those who opt to campaign for Brexit are in great trouble indeed.