Trump wants to break up Europe; Theresa May wants to help him. After May’s speech on Tuesday, the new defence line of the left and centre will be clear. We can and must own the Brexit decision, implement it without rancour, and then fight: to remain inside the single market, or seek substantial access to it.

But that also means implementing the express desire of the majority to end free movement. In a speech last week, Jeremy Corbyn recognised that freedom of movement, though a principle of the EU, is not also a principle of socialism. It has undermined social justice and must be modified. But how?

Let’s start with the outcome we are trying to achieve. For the left, it should be to minimise the economic break with Europe and restore popular consent for migration. Britain should apply to join the European Free Trade Area (EFTA) and, through it, to remain inside the European Economic Area (EEA). Those who say that means accepting complete freedom of movement are, not for the first time, misleading us.

Freedom of movement has always been a “qualified right” – not an absolute one: that is, constrained by national conditions. Plus Article 112 of the EEA treaty allows us to suspend freedom of movement, for an unspecified period and unilaterally, due to “serious economic, societal or environmental difficulties”. Well, we have a serious societal difficulty: we have lost consent for high inward migration, and we need to regain it.

We have to do something that meets the objections of low-paid workers to wage suppression, that kills the lie that migrants are responsible for the crisis of public services and that promotes integration as the overt goal of migration policy, not a side-effect.

First, we should take radical measures to deter the creation of low-wage businesses that can only exist because of an unlimited supply of temporary migrant labour. Corbyn has spelled these out in his speech, but I would go further: laws that discourage agency work and promote sector-level wage bargaining and union-only recruitment agreements.

Next, we should make a unilateral offer: to give the 3.2 million EU citizens currently in the UK the unconditional right to remain, either as British citizens, dual nationals or permanent residents. And we should give all who stay the right to vote in a general election, not just the local and devolved assembly elections as now.

Then we ask the EU for a 10-year, temporary suspension of free movement in order to make one small but effective tweak.

We redesign the National Insurance (NI) system so that the state can limit the supply of new NI numbers in one category: private sector job offers on less than a certain wage (for example £18,000 a year pro-rata).

The Department for Work and Pensions would monitor the specific job markets that rely on low-paid immigrant workers in real time and agree quotas for new NI numbers with, for example, the private social care or hotel sectors. Nobody coming from the EU to do a public-sector job would feel a difference. Ditto for the self-employed (though we need a crackdown on self-employment scams). People moving to skilled jobs in the private sector would feel little change. And all citizens of the EEA could still travel, retire, buy property or study freely as now.

But there could be no new NI numbers issued for people seeking agency work. Future migrants from the EU looking for low-paid jobs would find fewer of them. Logically, we could expect a few slave-driving employers to go out of business. Those who survive would have to offer fixed-term contracts, publish their wage rates and play by rules agreed between employers and the unions.

The move would not discriminate against any worker currently living in the UK. In fact, it is designed to improve the wages and conditions of everybody on low pay. More than that, it gives progressives an argument-starter in pubs and on doorsteps where they are fighting xenophobia. If it’s cheap labour and lack of integration you are worried about, our proposal sorts it – and gives us a chance to stay inside a market of 500 million people; a chance the Tories are prepared to throw away.

The move should not be unilateral: it should be the subject of a final Brexit agreement between the UK and the EU27 that is sold enthusiastically to electorates, both in Britain and the EU.

In Britain this move could be the weapon with which we separate the hard-core racists among Ukip and Tory voters from people primarily worried about wages, services and cohesion.

Having made the change, we should irrevocably scrap the Tories’ sub-100,000 a year migration target, reducing obstacles for employers and universities seeking to recruit beyond the EU. And we should design a new asylum system that makes the rights of the refugee paramount.

Corbyn’s speech prompted charges he was “pandering to racism”. This betrays a profound misunderstanding of what drives opposition to free movement among progressive, left-minded people in the communities where Labour is rooted. Free movement does not just suppress wage growth at the low end. It says to people with strong cultural traditions, a strong sense of place and community (sometimes all they have left from the industrial era) that “your past does not matter”. It promotes the ideal worker as a rootless person with no attachment to place or community, and with limited political rights; whose citizenship resides in their ability to work alone. In case you haven’t noticed, shouting, “Don’t be racist!” at the Labour voters who backed Brexit, isn’t working. That’s because most of them are not racist.

There is no point trying to appease hardened xenophobes. Like Michael Gove, they are mesmerised by the narrative of imperial white supremacy. But there is every point in trying to do something for people whose wages have been suppressed by free movement and the toughest anti-union laws in Europe. Labour needs to spell out its alternative to free movement loudly, proudly and fast.