I have found lots of information online about European pilots owning and flying N-reg aircraft using trusts set up in the US. I live in Europe, but I'm a US citizen, so must I also set up a trust because I don't live in the US? I cannot seem to find a definitive answer on this with references to supporting regulation.
As a US citizen you can own an N-reg aircraft directly, regardless of where you live. The regulations on aircraft registration are in 14 CFR Part 47. They say who can register an N-reg aircraft, but they don't say anything about where that person must be. 47.3(a)(1) simply says:
§47.3 Registration required.
(a) An aircraft may be registered under 49 U.S.C. 44103 only when the aircraft is not registered under the laws of a foreign country and is—
(1) Owned by a citizen of the United States;
The regulations are summarized in a more readable way on this FAA site. A trust allows a non-US citizen (or non-resident alien) to own an N-reg aircraft provided they have no more than 25% of the voting rights in the trust, per 47.7(c)(2)(iii).
No, as a US citizen you do not need to jump through any hoops to register an N-reg aircraft in your name, anywhere in the world. It doesn't matter where you live. I used to own an N-reg aircraft when I lived in Germany. It was registered to my German address. When I moved back to the US and brought it with me I submitted a change of address, and that was the end of that.
According to NBAA, trusts are used by non-US citizens to register aircraft with the FAA. Because you are a US citizen, it looks like you don't need a trust to maintain the FAA registration. Here's the article with references to the FARs: https://nbaa.org/flight-department-administration/aircraft-registration-transactions/owner-trusts/ . Here's an aviation trust company: https://aviationtrust.com/ . I have no financial interest in that company. They appeared higher up in my search.