The "real employee without registering an entity" problem has a pretty clean solution that a lot of companies still don't know about
Remote, Deel, Papaya Global all do it, and the EOR employs you in whatever country you're actually in, and the entity problem disappears because the EOR is already registered everywhere. It's not cheap but it's a lot cheaper than the compliance mess of getting it wrong. On the double-tax side I find that most countries have tax treaties with the US that prevent you from paying full tax twice on the same income, but the treaties are not automatic, you have to file correctly in both places and claim them. The 183-day rule is the rough framework that most countries use for tax residency, so if you're genuinely nomadding and not spending more than six months anywhere, you may not trigger residency in any of them. that sounds great until you realise the US taxes you on worldwide income regardless of where you live, so you're still filing a US return either way.
FEIE (Foreign Earned Income Exclusion) is the main tool for US citizens and lets you exclude a chunk of foreign-earned income from US tax, roughly $126k for 2024. It has requirements around physical presence or bona fide residence abroad that four-country years can actually satisfy pretty cleanly, but you need someone who actually does expat tax work to run the numbers, not a regular CPA who files W2s all day.
The four-countries-in-one-year situation is genuinely complicated and the honest answer is the generic advice on Reddit will get you part of the way there and then you'll need someone like Greenback or Bright!Tax to actually file correctly. the cost of getting it wrong is worse than the advisory fee.