A hedge fund can trade both but might be subject to regulatory requirements from both.
Nobody would want that.
Over my career I'd been both an RIA and CPO/CTA (not at the same time, of course).
While some "advisors" have gone their entire careers without a regulatory audit (hard to fathom that), I suffered 7. 3, from the SEC and 4, from CFTC. There were never any complaints against me, but because Denver was/is? a regional office and I was "small"... I think the regulators sent their rookies out to audit me for "experience". Didn't appreciate that at all.
Anecdotally... After an audit you get a letter citing everything they found which was not in compliance... some of which can seem petty. (Assuming of course that they didn't find anything
really bad so that they seized your records, locked your shop and hauled you away in handcuffs.) The one gig I recall is "not having full client information" on each customer's "account card" paperwork. Name, address, phone, etc. I had only the name.... the other info was in the client's folder in multiple places.
6 weeks after sending me the letter citing the above, they sent an agent out to follow up and make sure I'd put the rest of the info on each client's "account card". (They usually don't notify they're coming... they just show up.) The account card is an important document. It shows information about initial/subsequent investments and redemptions. Having "full customer info" on it isn't important... it's just there for regulator/auditor convenience.
At another time, I was a Principal in a small Broker-Dealer. We had 2 people whose full-time job was "compliance". Regulartors take their job very seriously... as would any gung-ho bureaucrat.