US corporations were previously prohibited from owning land or natural resources on the moon or other "celestial bodies."
The new rules, adopted in April 2020, say if an American company extracts resources on the moon or other celestial body, the company gets certain legal rights to transport, use and sell those resources.
This means corporations could control some limited territory in space -- where space resources were found or landed.
There are, however, some significant caveats. Any such space mining company would need a US federal permit. In addition, the resource extraction has to be done under the rules and obligations of the Outer Space Treaty.
The Outer Space Treaty is the main international treaty governing space exploration and use. Among other things, it prohibits national and private appropriation of space itself.
While the new US rules state this is permitted under US federal law, they cannot create international or universal rights -- such as to the territory in space.
The company mining the resources could use and benefit from the location to enable the space mining or transportation, such as by having temporary equipment there or landing spacecraft there.
Such a space "base" would not necessarily mean territory in space is fully controlled by the corporation, particularly under international law, depending on the nature and use of the location.