Because AQHA requires all horses to be registered with them in order to register their offspring, the only exception would be an AQHA/ jockey club horse registered as appendix.
The dual registered AQHA/APHA only applies to horses that are overo because tobiano is NOT part of AQHA. Any horse you see that is AQHA/APHA will have either AQHA parents or dual registered parents. Plus all of these horses must be DNA'd and parentage verified.
Just because an APHA horse is solid does not make it eligible for quarter horse papers. The part people have to understand is that splash overo, frame overo, and sabino are part of AQHA's lineage. Tobiano specifically is not. If you look at pedigrees, many tobianos go back to "unknown paint" very close up sometimes within 3-4 generations. AQHA as a breed does not, to my knowledge, have "unknown" listed near that close and they have strived to close the books and really become a breed. APHA wasn't founded until 22 years after AQHA. The associations closed books at different times and have had different rules regarding white requirements.
The dual registration started when AQHA rejected excessive white horses, so they obtained APHA papers. The lineage of the cropouts was still tracked and many of them did have DNA testing done. The registration business is about making money, so when enough of the membership was able to effect change, AQHA changed the white rules, and certain horses were allowed back in creating dual registration.
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Ok, makes sense... So in the long run, the only way to breed my filly and have her offspring be registered, she must be bred to APHA? How can horses qualify to be double registered?