When Theresa May visits Ottawa on Monday, she might have more to learn from Canadian prime minister Justin Trudeau than what socks to wear to the United Nations, or how to win a majority. Now that the UK has entered into the difficult early stages of negotiating its way out of the European Union, May and her government might want to take note while in North America of how another trade relationship works and how Canada, specifically, handles its role within it.

Donald Trump’s election put the spotlight on the North American Free Trade Agreement (Nafta), a deal between the US, Canada and Mexico that covers everything from avocados to lumber to Ski-Doos. Trump has made it clear that he believes Nafta to be “the worst trade deal ever signed”. The goal, as far as Trump is concerned? Regain control. As with similar promises from Brexiters, the impossibilities of such economic independence were totally ignored in favour of rhetoric. The unprecedented trade talks requested by both Trump and the Brexiters have arrived, and the way by which both Canada and the UK have approached negotiations is a study in contrasts – one that could be instructive for May.

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Whether Canada bargains at the Nafta table from a position of strength or weakness depends on who you ask and when, but the overall picture is that of an uneven partnership. The US claims it carries a trade deficit with Canada (it does not), but the overall Canadian economy depends more on America than the other way around. But this allows Canada to naturally land a key mental leap – one the Brexiters in May’s government should study. That is, while Canada has long accepted its lot as a junior partner in its relationship with the US, it also knows doing so doesn’t necessarily mean it’s not in a position of considerable power or influence. It all depends on the approach.

After the Trump administration prompted Nafta renegotiations, Canada pushed a charm offensive. It routed its diplomatic forays around the White House and down to the congressional and state level. As Vice News and Bloomberg first reported, Trudeau’s office dispatched as many as 18 cabinet ministers to lobby states directly and to remind them of the importance of Canada’s trade relationship in a more local context. Ministers held meetings, shook hands, and smiled for some Instagram photos.

Unlike the exchanges between the UK and the EU, Canada’s official verbal dust-ups with the US have been muted. As the renegotiations got under way, chief US negotiator Robert Lighthizer took direct aim at both Mexico and Canada, clarifying that Trump “is not interested in a mere tweaking of a few provisions and a couple of updated chapters”, but rather a rewrite. “We feel that Nafta has fundamentally failed many, many Americans and needs major improvement,” he said.

Canada’s foreign affairs minister, Chrystia Freeland, didn’t directly retaliate. Rather, her response was in keeping with the overall tone Canada adopted from the outset: measured, calm, and reminding of a mutually beneficial relationship. More Canadian, if you will. “We pursue trade, free and fair, knowing it is not a zero-sum game,” Freeland said, before pointing out that “that we are the biggest client of the United States”.

For their part, Justin Trudeau and Donald Trump share what is, at least in public, a congenial relationship. Trudeau is cautious to never sound like he’s attacking the president directly, and Trudeau’s inner circle has worked to establish friendly ties with his closest advisers. How much of it is genuine or merely for the cameras is immaterial; it’s in service of a greater goal. You catch more flies with maple syrup, I suppose.

There are genuine points of contest between Canada and the US when it comes to rewriting Nafta, and there will be disagreements in the course of its renegotiation. Nafta may even yet be destroyed. May’s Conservative government has an arguably tougher task with regards to the EU, but expecting Brussels to acquiesce, or to bend to Britain’s will, is fantastical thinking. It’s not going to happen – certainly not in a way in which the Brexiters’ lofty guarantees can be met.

With their vote to split from the EU, Britons perhaps unknowingly accepted a simultaneous fate of creating a lopsided economic relationship, with the UK dependent on the EU’s moves, rather than the other way around. In a way, Britons voted for the UK to become the Canada of Europe. Maybe it’s time it started acting like it.

• Colin Horgan is a former speechwriter for Justin Trudeau