I once flew an N regd Debonair in Alice Springs. (Pvt ops) After discussions with CASA and research on the FAA website, I was legally able to fly this acft in Oz. For memory, because the US and Oz are members of the Chicago Convention, and I held a PPL or higher, all is okay. And here is the great snag. CASA could not (were not allowed to ) ramp us, because it is an "Amercian piece of soil" and they would be trespassing - (that's straight from the CASA agent's mouth).
And there were probs with maintenance. 100 hrlys could be conducted by Oz LAMEs, but the 2 yearly check(or whatever) could only be conducted by Hawkers at BK.