Last fall in Bavaria, the Christian Social Union, regional ally of German Chancellor Angela Merkel’s center-right party, tried to steal some of the political clothing of the far-right Alternative for Germany, only to lose a significant chunk of its traditional voters. Last month, Spain’s center-right Popular Party forfeited more than half its parliamentary seats after trying to echo the themes of far-right populists. Britain’s Conservative Party faces a similar challenge if the Brexit Party emerges as a clear winner. And last week, Britain’s Justice secretary warned his fellow Tories to “avoid the temptation to be a populist party, which would narrow our base.”

The new Brexit Party is being projected as the main winner of Britain’s vote. And it embodies a challenge facing established center-right parties: Do we fight populists or join them? Some conservative parties in Europe have chosen to tack right. But Britain’s Euro-vote seems likely to provide new evidence this strategy isn’t working.

It may seem odd: Unless a Brexit deal materializes, Britain will vote in upcoming European Parliament elections. And the results will be very much worth watching, for what they’ll say about the unsettled politics of major Western democracies.

It’s shaping up as somewhere between surreal and farcical: Britain’s planned participation next week in elections for the legislature of the European Union – an alliance that the United Kingdom actually voted to leave in a 2016 referendum.

Yet unless Britain’s beleaguered prime minister, Theresa May, finds an 11th-hour compromise to get a Brexit deal through Parliament – avoiding the need to take part in the five-yearly EU vote – the results will be very much worth watching. That’s not just because of what they could mean for her Tory party, but also because of what they’ll say about the unsettled state of politics in a range of major Western democracies.

The group being projected as the main winner – the newly created Brexit Party of Nigel Farage, the anti-EU firebrand who galvanized the “leave campaign” in the 2016 referendum – is the latest embodiment of a challenge facing long-established center-right parties in Europe and in the United States as well. With nationalist, anti-immigrant, populist politicians gaining influence, “establishment” conservatives are having to ask themselves: Do we fight them or join them?

The response has varied from country to country. But as in the U.S., where congressional Republicans have been re-tailoring their policies and pronouncements to move into the slipstream of President Donald Trump’s “America First” populism, some conservative parties in Europe have chosen to tack in the direction of the far-right.

The significance of Britain’s Euro-vote is that it seems likely to provide new evidence that their strategy isn’t working, at least so far.

Last fall in Bavaria, the Christian Social Union, regional ally of German Chancellor Angela Merkel’s center-right party, tried to steal some of the political clothing of the far-right Alternative for Germany (AfD), only to lose a significant chunk of its traditional voters. Just last month, Spain’s main center-right political force, the Popular Party, forfeited more than half its parliamentary seats after trying to echo the themes of the rising, though still fringe, far-right populists of Vox.

For the true faithful of the AfD and Vox, their pitch clearly didn’t ring true. And the PP in Spain, like the CSU in Germany, prompted many longtime supporters either to look for less extreme-sounding options or to stay home.

Britain’s Conservative Party faces a similar challenge, both from Mr. Farage and from a vocal minority of anti-EU ideologues within its own ranks who want no part of Ms. May’s effort to find a form of Brexit that manages to retain valuable economic ties with the EU.

Under ordinary circumstances, the European Parliament elections needn’t affect the Tories’ domestic political fortunes. One reason is technical. The European Parliament vote is held under proportional representation. Elections for the U.K. Parliament use a “first-past-the-post” system. That means that a group like Farage’s Brexit Party can do really well in the EU vote, but still achieve, at best, third-place showings in Britain’s national election constituencies and end up without a single seat in Parliament.

Yet since the 2016 referendum, with the failure of Ms. May and Parliament to agree on a way actually to leave the EU, Brexit has crowded out almost every other issue from national debate.

If Britain does leave, the winning candidates may not even take their seats in the European Parliament. But especially if Mr. Farage’s party emerges as the clear winner, the results are bound to cause serious infighting within the parties that do less well.

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That’s especially true of the Tories, projected to suffer by far their worst showing ever in a European Parliament election. Ms. May is already under growing pressure by some of her most vocally anti-EU MPs to step down. They argue that, under a different leader, Britain could drive a better exit deal or, if that failed, simply walk away without one.

Still, not all Tory MPs agree. David Gauke, Justice secretary in Ms. May’s government, echoed her view last week that it was imperative to try to get her Brexit deal through Parliament before the Euro-vote. And perhaps with precedents like the Spanish and Bavarian elections in mind, he added a warning for fellow Tories, against drawing the wrong conclusion from a Farage victory. It was important, he said, to “avoid the temptation to be a populist party, which would narrow our base.”