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How can I legally and efficiently help my girlfriend build equity by helping with a mortgage?
There is no simple, legally reasonable, way for her to build equity by helping out with your mortgage, without her having a claim to your mortgage. The only 'equitable' thing she can do is rent from you.  If you want her to be building equity, have her start and fund a brokerage account for herself. If you have an affinity for real estate, have her buy REITs in said investment account.