Anglo-Arabians, Selle Français (or French Riding Horse), and French Trotters plus all other crossbreds can be qualified as AQPS. For racing purposes, each breed has its own studbook.
There are some registries that accept horses (and sometimes ponies and mules) of almost any breed or type for registration. Color is either the only criterion for registration or the primary criterion.
These are called color breeds, because unlike “true” horse breeds, there are few other physical requirements, nor is the stud book limited in any fashion.
Most warmblood registries recognize breeding stock from any other registry that is a member of the World Breeding Federation for Sport Horses, which is affiliated with the IOC-recognized International Federation for Equestrian Sports.