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The other survey, from the European Commission's Eurobarometer, also found an upswing in the share of European citizens who view the EU positively and have trust in it. Again, the upswing occurred in virtually every country.

What's going on? How did the EU turn its reputation around?

To some extent, Europeans may simply be realizing that the grass isn't actually greener on the other side the other side being, in this case, life outside the European Union.

Britain's upcoming exit has led to political chaos and economic uncertainty, not to mention sagging consumer confidence and departing jobs. Tens of thousands of jobs may leave London's financial sector alone.

The same Pew survey found that majorities of nearly every country say Brexit will be bad for both the EU and Britain. Even a plurality of Brits believe Brexit will end badly for them. (Greece, which was threatening to "Grexit" the euro zone before departure portmanteaus were cool, is the only surveyed country in which a plurality believes Britain will be better off.)

Perhaps other EU members have watched Britain's isolationist dysfunction and started to better appreciate the European project, even with its many flaws.

Not just coincidentally, in no country that Pew surveyed did a majority of respondents say they want to leave the European Union. This finding jibes with other recent polls.

Nonetheless, even though they don't want to leave, in nearly all of the countries at least half of respondents still want to hold a referendum to vote on whether to leave.

This may seem peculiar, given that Britain got such an unwelcome surprise when it held its own referendum. But this desire to hold a vote may reflect frustration with the lack of a say in what happens in Strasbourg (and Brussels, Luxembourg and Frankfurt, where other major EU business gets done). A referendum could be viewed as a way to gain more leverage over EU officials, even if the vote is really a bluff.

"People think that voting will empower them," says Luigi Zingales, a University of Chicago professor who has studied economic and public opinion trends in the EU. "Most Europeans are happy with the idea of some form of European integration and the common market. They just want more voice in the process."

Zingales also argues that a force bigger than Brexit may be more important in reviving the EU's reputation: the fact that finally, a decade after the global financial crisis struck, so many European economies are actually improving.

Zingales notes that in the Pew data, only his home country of Italy hasn't started feeling more warmly toward the EU. Italy also happens to be the only surveyed country whose citizens are more pessimistic about their economy today than they were a year ago.

"When things go poorly, you blame everybody: your government, the EU government, probably also the United Nations," he says. "When things go well, maybe you're now sort of OK with everything."

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Catherine Rampell: The newly popular European Union - Salt Lake Tribune