The family moved to southern Florida in 2006 to focus on tennis full time. As other children went off on the school bus, the sisters trained most of the day on the Pembroke Pines public courts and were home-schooled at night. The girls grew in strength and talent, and in time, Tamaki decided they should meet their Japanese family, from whom she had been largely estranged for nearly 15 years. And so, when Naomi was about 11, she and her sister visited their grandparents in Japan. It wasn’t as joyful a homecoming as Tamaki might have hoped. Her parents took an interest in the girls, she says, but ridiculed their regime of home schooling and tennis training. Tennis was a hobby, they grumbled, not a profession.

Back in Florida, the girls skipped many of the usual circuit of junior tournaments and, eventually, started competing against older players on the pro satellite tours, just as the Williams sisters had done. With a growth spurt in her early teens, Naomi soon towered over Mari. Video clips of the girls’ matches and training began circulating among coaches and agents, but neither sister had an impressive junior ranking or much tournament experience. The United States Tennis Association showed little interest in helping them develop. Rather than vie for support with hundreds of other talented young players in America, Francois made a pivotal decision: His daughters, from age 13, would play for Japan, the nation they left behind nearly a decade earlier.

“My dad thought that since I grew up around my mom and I have a lot of Japanese relatives ... I don’t know. ...” says Osaka, letting the sentence drift off. Despite growing up in United States, with all the cultural references of a typical American youth, she told me: “I don’t necessarily feel like I’m American. I wouldn’t know what that feels like.” Her sister speaks almost fluent Japanese, but Osaka’s grasp on the language is more tenuous. “I don’t know if you guys know this, but I can understand most Japanese, and I speak when I want to,” she tweeted earlier this year, adding: “That applies to my family and friends.” She says she is too shy — and too much of a perfectionist — to speak the language publicly. Her reluctance can create awkward moments at news conferences, with Japanese reporters asking questions that she answers in English.

The decision to play for Japan has had major repercussions in Osaka’s life, from the way she is perceived in Japan and the United States to the size of the endorsement contracts she can now command as a top Japanese athlete ahead of the 2020 Tokyo Olympics. Though some in the tennis world wondered whether the decision was influenced by commercial prospects — the Japanese star Kei Nishikori’s massive endorsements were no secret — the family insists that the girls were too young and unproven for that to be a factor. The Japan Tennis Association, facing a drought of top female players, offered them an opportunity. But for Tamaki and Francois, who spent many years in Japan himself, it was natural for the girls to play in the country where they were born, even if the parent’s own memories of the place were tinged with anger and regret.

The bearded man on stilts bellowed into a microphone, exhorting the crowd to chant her name: “Na-omi! Na-omi!” The hype machine was revving up for a World Team Tennis match, a nonranking format designed to turn the sport into popcorn-eating entertainment. In the stands of the Washington arena on this July evening, cheerleaders in slit white miniskirts and tight red tops swung their pompoms while young men paraded around twirling enormous cutout posters of Osaka’s likeness. Above the tennis court — a Mondrian-like matrix of green, purple, blue and red boxes — four giant screens broadcast images of its headline act on the court below.

For the women’s singles match, one first-to-five-game set, Osaka was pitted against Taylor Townsend, an American ranked 44 places below her. It was expected to be a comfortable win for Osaka and her team, the Washington Kastles, but the circus atmosphere — and the pressure to win every game for a team that had brought her in just for this event, to help propel a playoff run — seemed to throw her off. Her first three service returns careered out of the court. As the errors piled up, a sore right calf got worse. At one point, the Kastles’ announcer pumped up the crowd. “Refuse to lose!” he yelled. “Get ’em up, get ’em up, get ’em UP!” Osaka still lost in a tiebreaker, dumping the final ball in the net and trudging off the court.

Two hours later, Osaka sat courtside, stone-faced, as her Kastles teammates pushed the overall match into a deciding doubles tiebreaker. Mari had been waiting to go with her to the Beyoncé and Jay Z concert in Maryland that night. The sisters had been planning this outing for months, but the W.T.T. match was blowing right through the opening act. Earlier this year, when Osaka thought she might miss Beyoncé’s tour, she tweeted: “Tell me why Beyoncé decides to have a concert in Miami at the same time as the US Open. I’m legit gonna cry.”