One is an international pop star who just went to space. The other is a clothing designer. A court in Australia will decide which gets to claim their name.

Katie Perry, an Australian designer with a namesake fashion label, has for years asserted that Katheryn Hudson — the international superstar known to most of the world as Katy Perry — has infringed on her trademark.

Her case made its way through Australian courts for more than five years, with mixed results. The designer won in 2023, but an appeals court reversed that last year, seeming to settle the issue.

Now, Australia’s High Court — the highest court in the land — is revisiting the fight, suggesting that the appeals court may have relied on a faulty premise about fame that could threaten Australian brands.

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“This is a tale of two women, two teenage dreams and one name,” one Australian judge said in 2023. But the case has wider implications and raises questions about reputation and the many ways celebrities make money.

The issue between the two Perrys first arose around 2009, the year after the American singer released her hit debut single, “I Kissed a Girl.” The star’s representatives say that they reached out to the designer to make a deal over use of their similar sobriquets, and that she declined. She says she never received an offer.

Katie Perry, the designer, says she launched her label in 2006, started working on it full time in 2007, applied for one of her trademarks in 2008 and registered it in 2009. That was all, she claims, before the singer made it big in Australia.

A decade later, she sued Katy Perry, saying that clothing sales on one of the singer’s tours through Australia infringed on the designer’s trademark.

The designer won her initial case. A judge said the American celebrity had no reputation in clothing when the Australian applied for a trademark. That meant the star had no superior claim to her stage name when it came to clothing sales, and the judge found that some items sold on her Australian tour had infringed on the designer’s mark.

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But last year, an appeals court overturned the decision. And, in a surprise twist, it went a step further: The judges also said the Katie Perry trademark should never have been granted, and they ordered the designer’s mark canceled.

The pop star already had a reputation when the designer applied for her trademark, they said, making it likely the celebrity would expand into merchandise and that the Australian label could confuse consumers.

In choosing to revisit the case, the High Court expressed concern that the appeals court’s seemingly anodyne decision could, in fact, have set a troubling precedent.

What follows from the appeals court’s logic “is far more significant than seems to have been appreciated,” High Court Justice Jayne Jagot said at a hearing last week, shortly before she and her colleagues granted the designer’s request to hear her appeal, something that they do in only about 10 percent of cases presented to them, experts say.

The justices questioned the appeals court’s reasoning, suggesting it had relied on a potentially problematic proposition that entertainers would expand their brands. They mused about how widely it might apply and whether this approach also required an ability to peer into the future.

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“If you are sufficiently famous, the capacity is to monetize in all kinds of directions, not just clothing: It is whiskey, wine, makeup,” Justice Jagot said. “It becomes difficult to think of a commercial activity that would not be covered.” She added that a celebrity could expand even into medical or dental services, like “Katy Perry Invisalign, or whatever.”

Justice Simon Steward agreed and said that there also seemed be an implied assessment that the star’s brand would continue to succeed, expand and monetize indefinitely, which would have been difficult to ascertain. “How could that have been known that it would continue to be successful back then?” he asked the celebrity’s representative, Matthew Darke.

Mr. Darke insisted — again and again — that there was no legal question with broader consequences. He denied that the appeals court had made mistakes. “Perfectly appropriate for the court to reason as it did, and there is no error,” he said.

The justices were not convinced and called for yet another round in this long-running dispute.

In its deliberations, the High Court will have to decide just what Katy Perry was famous for around 2008 and whether it mattered if she had a reputation in clothing or was just well-known. The decision will determine how similar cases are litigated and decided.

And similar cases do arise.

For example, Burger King operates in Australia as Hungry Jack’s because another restaurant already owned the Burger King copyright when the franchise first opened in the country.

Gregory Pieris, an intellectual property partner at K&L Gates in Melbourne, Australia, said there were “many instances” where someone in Australia tried to register a trademark identical to that of an international brand. What happens next with Katy Perry and Katie Perry will shape future such cases, Mr. Pieris noted.