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No they are not "dating", but Italy has shown a lot of affection towards Germany throughout the series, and especially in the Buon San Valentino strips, Germany shows some attraction towards Italy.

Himaruya has also made a comic about Germany trying to get Italy's attention in a romantic way. He went as far as buying Italy a ring and trying to give it to him during a dinner date, in which their waiter takes it the worst way possible and Italy rejects Germany.