That is not how I see § 61.3 Requirement for certificates, ratings, and authorizations.(a)Required pilot certificate for operating a civil aircraft of the United States. No
person may serve as a required pilot flight
crewmemberof a
civil aircraft of the
United States, unless that person:
(1) Has in the
person's physical possession or readily accessible in the
aircraft when exercising the privileges of that pilot certificate or authorization -
(i) A pilot certificate issued under this part and in accordance with
§ 61.19;
(ii) A special purpose pilot authorization issued under
§ 61.77;
(iii) A temporary certificate issued under
§ 61.17;
(iv) A document conveying temporary authority to exercise certificate privileges issued by the Airmen Certification Branch under
§ 61.29(e);
(v) When engaged in a flight operation within the
United States for a part 119 certificate holder authorized to conduct operations under part 121 or 135 of this chapter, a temporary document provided by that certificate holder under an approved certificate verification plan;
(vi) When engaged in a flight operation within the
United States for a fractional ownership program manager authorized to conduct operations under part 91, subpart K, of this chapter, a temporary document provided by that program manager under an approved certificate verification plan; or
(vii) When operating an
aircraft within a foreign country, a pilot license issued by that country may be used.
Part .1 says a US cert
Part vii says a local cert MAY be used
ie either a US or an Australian certificate.