In the eyes of the world, Brexit and the Trump presidency are two sides of the same coin. Their triumphs in 2016 are seen as part of the same world-wide antiestablishment, anti-globalization rebellion.

Some Brexiters reject any link, arguing that Brexit was a vote in favor of free trade in contrast with President Donald Trump’s protectionism. Yet these Brexiters may have more in common with Mr. Trump than they care to admit.

That...