SO, you might ask, do other countries have bases on US soil?
I use to go in/out of Dulles a lot and there is a building marked, Luftewaffe. No, I don't mean Lufthansa, but German Air Force. Now they don't actually do operations there, it's more of a parts procurement facility, since they fly some US made planes and I bet we have some German parts in ours as well. But it was always humorous to see the German flag flying over the facility.
Now I did do some checking and there are German presence at other bases, but the above is the only formal "base" on US soil, not just having training units or liaison officers at our bases.