It would be a part 91 operation.
If I'm interpreting this FAR correctly:
§ 47.3 Registration required.
(a) An aircraft may be registered under 49 U.S.C. 44103 only when the aircraft is—
(1) Not registered under the laws of a foreign country and is owned by—
(i) A citizen of the United States;
(ii) An individual citizen of a foreign country lawfully admitted for permanent residence in the United States; or
(iii) A corporation not a citizen of the United States when the corporation is organized and doing business under the laws of the United States or a State, and the aircraft is based and primarily used in the United States;