BERLIN—In 2011, German tech company Rocket Internet SE spotted a Swedish web company delivering ready-to-cook meals—recipes and ingredients packed in a box. So Rocket did what it does best: It launched a copycat business in other countries.

Its knockoff, HelloFresh, boomed on three continents, including in the U.S., where it competes against other imitators such as Blue Apron and Plated. Then trouble hit.

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