It's illegal (one of the missions of US Secret Service is to ensure that there are no other currencies used in the US, and they have cracked down e.g. on Las Vegas casino tokens being used in surrounding restaurants to pay for drinks).
I meant for non-US banks. What stops "London" from setting up its own USD clearinghouse and having non-US banks start using that? Simple pressure from the US?
Possibly. Basically, the US can force the banks to do what they want, because the ability to retail within the US is worth so much. So either they, or the banks they work with, could be pressured to comply.