When Theresa May surveys the political obstacles ahead of her, the figure of Jeremy Corbyn does not loom large. A prime minister only has time to worry about a limited number of hazards and withdrawal from the European Union brings a unique hierarchy of problems. The Labour party ranks well below Tory backbenchers, continental diplomacy, the supreme court and a Europhile civil service. Were it not for the weekly requirement to answer questions from the opposition leader in the Commons, May might forget about him altogether.

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Fear of irrelevance provoked Corbyn into a flurry of activity on Tuesday – a dense schedule of broadcast interviews in the morning and a speech in the afternoon, all choreographed to monopolise the news cycle. That mission was accomplished. Anyone chancing across an afternoon bulletin would learn that the Labour leader thinks there should be some kind of cap on wages (but not how high the ceiling on earnings should be). Most would also glean that Corbyn had said something about immigration and Brexit – perhaps that Labour is no longer “wedded” to free movement of people but rejects the idea that numbers must come down.

The dearth of detail and confusion about what had actually been announced provoked ridicule in Westminster, but Corbyn’s team are relaxed about that. Their strategic plan for the new year factors in certainty that anything the Labour leader says will be savaged by pundits and disowned by some of his own MPs. The theory is that Corbyn can pull off a kind of jujitsu on his opponents, deflecting their own power against them; recasting the ferocity of an attack as proof of the target’s virtue.

If the combined forces of a corrupt establishment are so hostile to the Labour leader, he must be doing something right, or so the argument goes. And if the product of a day’s media frenzy is people arguing about something Corbyn has planted on the agenda – grotesque disparities between the pay of bosses and their staff, for example – it counts as terrain seized from the enemy. Better to be pilloried than ignored.

This concept flows from observation of the Vote Leave campaign and Donald Trump’s White House bid, both of which succeeded by making a virtue of notoriety. Both were denounced for lurching wildly beyond the bounds of acceptable argument. Both times the attack mistakenly presumed public respect for political orthodoxies and protocols as policed by elite institutions and experts. Corbyn’s strategists posit that this dynamic can be replicated to generate anti-establishment momentum behind radical left causes – chiefly the taming of megalithic corporate power, which many Conservatives also identify as an incoming populist tide.

There is a difference between seeing the wave and surfing it. There are many problems with the idea that Corbyn might replicate a Trumpesque subversion of conventional political wisdom. One is stylistic. Setting policy aside, Corbyn’s triumph in Labour leadership contests – a model that he would like to scale up for a wider audience – depended on a modest persona. He was the accidental hero who plucked the sword from the stone that had entombed Blairism. Admirers of Corbyn’s mild manners feel protective of him when he is attacked. They rally to his meekness, which is a different impulse to nationalistic admiration for swaggering braggarts.

The bigger problem is that Britain has already cast its vote for radical change and the verdict was Brexit. It is too late to tell people they boarded the wrong train, ask them to unload their anger and point them to a different platform. In fairness to Corbyn, his speech on Tuesday recognised this point. He tried to position Labour as a party that can be enthusiastic about Brexit as long as the terms are unlike those dictated by Tory libertarians, with their axe poised over employment protections, and Ukip nativists beating the drum for a lower foreigner count.

They are instinctive Eurosceptics, albeit coming at it from opposite ends of the ideological spectrum

The hard part is describing a left-friendly Brexit in terms that fire the imagination. Outside the EU it would be easier to push the state into parts of the economy where government subsidy is currently prohibited. But that is the stuff of theoretical, anti-capitalist Euroscepticism, which comes instinctively to Corbyn but doesn’t cut through into the bigger debates about openness to the world and what should be sacrificed for border control. That is the question that will dominate politics in the coming years. When Corbyn is pressed on it, he withdraws into vagueness.

May is no model of clarity, although she hints often enough that a Brexit without lower immigration is unworthy of the name. Her reticence is partly tactical. She knows that any statement of intent is a hostage to fortune in the coming negotiations. But stubbornness plays a part too. May wanted to be prime minister long before she was obliged to take Britain out of the EU and she is determined that Brexit will not define her. She has a doctrine of 21st-century Conservatism, based on greater government activism in pursuit of social mobility, which she is expounding in instalments. On that theme she can speak with fluency and moral urgency.

In that respect, she and Corbyn are oddly alike. They are instinctive Eurosceptics, albeit coming at it from opposite ends of the ideological spectrum. But Euroscepticism was not a passion for either of them. They half-heartedly campaigned to remain in the EU, more from tactical calculation than conviction: Corbyn because his party demanded it, May because she thought she was backing the winning side. They are now forced to make Brexit their defining purpose, having spent decades in politics with other motives and goals in mind. They both feel that the referendum result was the manifestation of some other discontent, which they interpret – as most people do – as proof of things they had believed long before last year’s vote.

Neither May nor Corbyn has a problem with leaving the EU; but neither of them is animated by the process, still less excited by it. Both know they must walk the walkout of the EU, but both would rather talk about something else. On Brexit, Corbyn is barely an adversary to May at all. When it comes to avoiding the subject, they are on the same team.