The Outer Space Treaty was only meant to prevent governments from claiming ownership of celestial bodies. The SPACE Act echoes that:

>It is the sense of Congress that by the enactment of this Act, the United States does not thereby assert sovereignty or sovereign or exclusive rights or jurisdiction over, or the ownership of, any celestial body

But practically speaking there's also not much the US can do in that scenario if Russia or China ignores US sovereignty over Mars and sets up their own base. The US's only recourse would be to start a war, and would that be worth it? Sovereignty is more of a practical concept than a legal one: you own what you are both able and willing to defend.

See also the Russian bases in the Australian Antarctic territory. Australia asserts sovereignty over a third of Antarctica, Russia ignores it and sets up bases anyway. Australia has neither the ability nor the desire to fight Russia, so we studiously ignore it.

Australia definitely has the ability to prevent Russia from setting up bases in Antarctica. It more like that they don't intend to do that because all their claims of sovereignty over Antarctica is a joke.