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.

If you file a DMCA notice and the infringing party files a DMCA counter-notice, would you have to register the copyright before you could sue for infringement?

I believe the answer is yes, you need to first register before filing suit. If the offending work continues to be published, you can register and then sue.

An exception is foreign works, whose owners can file a suit without first registering, since treaty obligations don't allow the U.S. to subject foreign holders to "formalities". However foreign holders are limited to recovering "actual damages" in such suits, and need to register to be eligible for per-copy statutory damages.