Carlyle, the world’s second-largest private equity manager, is known in the U.S. for buying household names for tens of billions of dollars. In Tokyo, it often goes for family-owned companies with enterprise values of less than $300 million. Tamotsu Adachi, managing director and co-representative for Japan, says he’s sticking with this approach after finishing raising 119.5 billion yen ($996 million) for the latest fund last month. He says health-care and consumer companies are...