At the beginning of Channel 4’s drama Brexit: The Uncivil War, the ex-Tory Ukip MP Douglas Carswell tells Matthew Elliott and Dominic Cummings, newly appointed to run Vote Leave, that they will offer a “respectable alternative” to the “rightwing thugs” Nigel Farage and Arron Banks. Cummings, the “strategist” (and anti-hero of this semi-biopic), soon returns to the theme: the Conservative-led and soon to be officially recognised campaign “needs to be respectable”, appealing beyond Ukip’s base if it is to win 50% plus one and the referendum. Elliott drums the point home: “We need to get them [Farage and Banks’ Leave.EU] to do the heavy lifting on the migration stuff – then we keep our hands clean.”

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However Cummings has a problem. The EU’s supranational institutions, demonised by Brexiters, are “too complicated, too remote” to be the focus of the campaign. It needs a “simple message repeated over and over and over”, if it is to appeal to the “three million extra voters that the other side have no idea exist” who Cummings’ new friend Zack Massingham, of the shadowy AggregateIQ, promises can tip Vote Leave over the line.

What the programme doesn’t tell us is that they are habitual non-voters, the sort of people who have almost zero interest in politics – and are quite likely to include racists. To “appeal to their hearts”, Vote Leave needs a message very like that of the despised Farage and Banks. “£350m and Turkey” shouts Cummings, standing on a table, to his staff. Or to be more accurate, £350m, Turkey and the NHS.

What the film doesn’t tell us is how low he went to push this. It shouldn’t be a secret, since Vote Leave repeated the same election broadcast (now pulled from YouTube) on all terrestrial TV channels throughout the last 30 days of the campaign. Following lurid graphics representing the threat of 76 million Turks joining the EU and coming to the UK, it climaxed with a split screen showing (staying in the EU) a surly foreign man elbowing a tearful elderly white woman out of the queue in A&E, while (leaving the EU) the woman is contentedly treated without having to wait. It was a homage to Enoch Powell, whose 1968 speech highlighted a fearful old white woman living in a street taken over by “negroes”.

Cummings’ 1bn Facebook ads picked up the broadcast’s themes and graphics, but were “micro-targeted”. Less racist voters got pictures of Boris Johnson (“I’m pro-immigration, but above all I’m pro-controlled immigration”), while the “3 million” got ads shouting “5.23 MILLION MORE IMMIGRANTS ARE MOVING TO THE UK! GOOD NEWS???” and when they clicked “No” were bombarded with scores of variations on the theme. It was the same story with leaflets, whistleblower Shahmir Sanni says: “The campaign was always talking about immigration. The most proud moment for many of Vote Leave’s staff was how well the Turkey leaflet did.”

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Leave voters didn’t like being accused of racism: “We can’t say nothing now without that coming up,” complains a focus group member in the film. Indeed the point isn’t that anti-immigrant voters were racist, but that Vote Leave assumed they were. The ads laboured how poor Turks were – they didn’t need to say that they were Muslim and foreign, but these were the buttons they aimed to push. The racism was Vote Leave’s, and it was easily turned against the other 3 million, the Europeans in the UK. “Project Hate”, Sadiq Khan called it.

The programme shows Cummings demanding “complete independence”, with Vote Leave’s figureheads Johnson, Michael Gove and Labour’s Gisela Stuart (who chaperoned Johnson on the notorious bus) barely understanding what he was doing. “That is just the actual population of Turkey,” replies Johnson, when a voter throws “70 million Turks” back at him; “I thought we weren’t pushing immigration,” says Gove. Only a game Penny Mordaunt actually tells Andrew Marr: “One million may come here from Turkey in the next year.”

It’s all too convenient. The drama lets Cummings blame Banks and Farage, while he is the fall-guy for Johnson and Gove, who commissioned him to run this shameful campaign. They can’t tell us that neither they nor any of their staff ever watched the election broadcast, picked up any of the leaflets or saw its Facebook ads.

Vote Leave learned from Farage that you couldn’t win Brexit without immigration, and brought racism to the centre of British politics. Now Theresa May is selling her deal the same way.

• Martin Shaw is a political sociologist with links to Roehampton and Sussex universities and the Institut Barcelona d’Estudis Internacionals