TOKYO—Over the past few years, as many people rushed to trade in their old phones for smartphones, Japan's philanderers have remained faithful to one particular brand: Fujitsu Ltd.'s older "F-Series" phones, which feature some attractive stealth privacy features.

The aging flip-phone—nicknamed the "uwaki keitai" or "infidelity phone"—owes its enduring popularity to customers who don't believe newer smartphones are as discreet at hiding their illicit romances.

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