Tri-color isn't a term commonly used in poodle circles. Try searching for parti phantom instead. What is your reason for wanting a full registration? Most reputable breeders are very protective of their dogs; they are concerned their puppies will end up with a doodle breeder or puppy mill.
Go to this site, the United Poodle Association. They register and show poodles that are parti, sables, phantoms, abstracts, tuxedos, and brindles.
To see those different types, to to site, use the top horizontal drop down menu > Poodle Info > Poodle Coat History.
To see some of the show dog pics, use the top horizontal drop down menu > Wall of Honor.
To see members, many who are excellent breeders, go to Membership > Members.
In case you're looking for a Merle, you won't find them shown or accepted for membership, because "Merle is NOT a naturally occurring poodle color and indicates another breed was used and papers falsified in the pedigree. It is considered a mixed breed, NOT a purebred poodle. It is a disqualification in UKC and not allowed registration."
In poodley-looking Merles, someone snuck in an Australian Sheppard or Collie way back in the ancestry to get the eye color and tricolor coat, but lied about the parentage and falsified records. A bad gene mutation called MDR1 also snuck in with many of these dogs too. This causes extreme sensitivity to a variety of common medications that are fatal, see that link for the list.
On top of all that, double Merles are more likely to have deafness and other problems. A surprising number of Merle breeders don't know or don't care about understanding basic genetics to at least try to avoid this, so these are other reasons to be aware of.
Anyway, good luck in finding a pretty poodle with a pleasing color pattern; many listed at the United Poodle Association.
Keep in mind that the breed standard for poodles specifies solid colors only. So anyone who purposely breeds for anything other than solid colors is not breeding to the standard. Certainly there are people who do breed for non-standard colors who also do genetic testing and who try to breed dogs who conform to the standard in every way except color. However, there are, sadly, people who purposely breed non-standard colors and advertise the puppies as "rare".
I just did a google search for "tri-color poodle" - the results I got looked like a list of the worst possible breeders! The very worst was someone who advertised that they would sell one puppy for $499 and a second one for just $299! That's appalling!
Vita gave you some really good suggestions for your search.