In this case, though, copyright infringement is already in principle illegal whether a work has been registered or not; the registration requirement is just a procedural requirement that bars collecting damages, not something that makes the action legal or illegal.

However U.S. copyright law does have a statutory limit on how retroactive registrations can be. If you file a registration within 90 days of first publication, the registration is retroactive to the date of publication; but if you file more than 90 days from publication, registration is only effective from the date of registration.

Fwiw, even with unregistered works you have some options available to you, just not damages. For example, you can file a DMCA takedown notice.