Hollywood actor Richard Gere, French film star Alain Delon and Russia’s figure skating sensation Alina Zagitova have one thing in common: They adore Japan’s Akita dogs. And they’re not alone. In recent years, foreign ownership of one of Japan’s most famous indigenous breeds has skyrocketed, outstripping domestic demand for the fluffy, perky-eared pooches. Earlier this year, the trend hit the headlines when Zagitova proclaimed her love for Akitas after spotting them while training in Japan, and local officials pledged to get one to her. Her affection for the cute canines comes as no surprise to veteran breeder Osamu Yamaguchi, 64,...