A key detail you seem to have missed throughout the seasons...If you recall, Tex also became a Ghost like Church back in Season 2. We know Church wasn't actually a Ghost, but an AI...
Throughout Seasons 8-10, it's explained that Director Church's memory of Allison was so strong that when creating the Alpha AI from his mind, a separate whole AI was spontaneously created: Beta, Agent Texas. It gave him the idea for the entire AI fragment experiment in which they tortured the Alpha to force personality fragmentation and used an alien Engineer (actually their name, in the Halo series) to fix fragmentation by siphoning the fragment into its own memory unit.
I know you've already finished the series, but maybe that helps clarify things if you're still confused.