ATHENS—On the Acropolis hill, hundreds of tourists braved the Greek sun on Sunday as they walked toward the Parthenon temple, pausing in awe at the ancient marble ruin, many of them oblivious to the different kind of ruin threatening modern Greece.

But Greece’s unexpected call for a July 5 referendum on the economic policies creditors want—which could push the country into bankruptcy and out of the euro—is catching up with tourists.

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