The Labour party is not in charge of Brexit, but it has nevertheless impaled itself on a dilemma about post-Brexit immigration. It can’t escape because it doesn’t know what to say. The uncertainty is understandable – because Brexit has sharpened the choices about migration policy – and at the same time bizarre, because the issue ought not to be causing the kind of trouble it undoubtedly is.

On one side of the argument, Jeremy Corbyn and John McDonnell obstinately make a virtue of refusing to talk about immigration controls. They prefer to focus on labour market regulation. “It is not an objective to reduce immigration,” a Corbyn spokesman famously said this autumn. These are “fictitious debates”, McDonnell said this week.

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On the other side, most Labour MPs in seats that voted to leave in June think toughening controls is both correct policy and an unavoidable response after the referendum. Freedom of movement is “no longer an option”, Andy Burnham said this week. “If you are asking me should we carry on with free movement as was,” Ed Miliband said in a recent interview, “I don’t think we should.”

This is a profound disagreement and some Labour MPs have lurched too far too fast in either direction. It hasn’t quite come to a head yet because, in the aftermath of Corbyn’s re-election, neither side of the party is keen to recommence the internal battles. But the spectacle of the main opposition party putting forward two irreconcilable policies – with frontbenchers such as Keir Starmer and Diane Abbott taking different positions on the same day – is unlikely to survive prolonged contact with the electorate, as the byelection disasters in Richmond Park and Sleaford showed.

Yet the disagreement is not a new one, and Labour has succeeded in managing it before. Back in the summer of 1965, Harold Wilson’s Labour government published a radically restrictive white paper on immigration from the British commonwealth that shocked even cabinet ministers. “This has been one of the most difficult and unpleasant jobs the government has had to do,” the housing minister Richard Crossman wrote in his diaries. “We have become illiberal,” he mourned. “This will confirm the feeling that ours is not a socialist government.”

Nevertheless Crossman was absolutely sure that the controls were necessary. “I am convinced that if we hadn’t done all this we would have been faced with certain electoral defeat in the West Midlands and the south-east,” he went on. “Politically, fear of immigration is the most powerful undertow today … We felt we had to out-trump the Tories by doing what they would have done … I fear we were right.” Antisemitism and racism were endemic in Britain, Crossman suspected. “One has to deal with them by controlling immigration when it gets beyond a certain level.”

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High-minded it isn’t. But those words echo today because the essence of the argument in which Crossman’s generation participated – hard times, more migrants, native resentments, press and public prejudice, liberal principles under challenge, electoral defeats – has not altered all that much. Yet just as the Wilson Labour party was right to grasp the issue, though it could have grasped it far better, so the Corbyn party needs to grasp it in an equivalent way too. Granted, much has changed since 1965. Migration today divides about equally between those from the EU coming under freedom of movement, and non-EU migrants who require permission. Nor do all migrants come to settle. Half of the overall total come to Britain to work, with a quarter coming to study.

The biggest change is the sheer numbers. In the early 1960s, the entire population of all black and Asian migrants in the UK was 336,000. Nowadays, the UK has upwards of 600,000 people immigrating each year, with about half that number moving abroad. Although voters exaggerate the scale of immigration – they think 25% of the population are migrants when the true figure is 13% – these are big numbers with big impacts.

The other unignorable change is Brexit itself. There are two reasons for this. The first is that the Brexit vote was clearly to some degree a plebiscite about immigration. The second is that, by voting to leave the EU, Britain has to decide whether to seek an association with a single market whose founding principles include free movement of labour.

Yet the political perils for Labour are very familiar. After the issue of Brexit itself, voters think immigration is the most important question facing the country. But Labour’s poll ratings on immigration are terrible. Only 11% of voters in the most recent YouGov poll think Labour is the best party on immigration, with only 29% of Labour voters from the 2015 election – which Labour lost badly – agreeing. A mere 5% of leave voters think Labour is best on immigration.

If Labour’s priority is to re-secure its core voters on the issue, that is a very bad place to start from. Latest research by Chris Hanretty of the University of East Anglia suggests that 64% of Labour’s 232 parliamentary seats voted leave in June. So if it is correct that immigration control was the decisive issue in the leave win, Labour MPs are right to demand that, at the very least, their party says something about immigration that engages with that stark reality.

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There’s a framework for this. For most of the last half-century Labour’s policy was that managed migration made community integration and mutual trust possible. The policy had periods of success and failure. It was too repressive in the 1970s and it was too insouciant in the early 2000s, after EU enlargement. But until recently it worked reasonably, and won electoral consent.

Labour’s challenge in 2017 is to renew that same approach around the realities of Brexit. It won’t be easy, and it will involve compromises of the sort that Crossman expressed half a century ago. The policy cannot be based solely on liberal principles. But it cannot be based solely on ignoring those principles either.

It has to place intervention in the labour market to ensure fair treatment, alongside an unsentimental approach to immigration control. If Labour’s factions can at least agree to start from there, there’s just a chance that enough of the rest will fall into place.