It doesn’t matter if there isn’t a written lease. Its still very much a rental arrangement.
That’s sorta the issue. You shouldn’t treat your SO as a tenant.
I would hope you treat your SO as an equal partner, but that also means healthy boundaries equal to where the relationship is at the time. If one doesn’t pay rent, but pays toward the mortgage, and you break up instead of getting married, do you expect the home owner partner to cut the other partner a check to cash them out of their “equity”? How is that fair to the homeowner?
I’m arguing non-homeowner had zero risk and should have zero equity.
The non-homeowner put zero money down for the purchase, they put none of their credit at risk, they took on no liability for the property, and so far there’s no mention of their obligation to pay for upkeep and repairs. Doing those things are the requirements of home ownership while the benefit is the equity. The non-homeowner simply hasn’t done the things to be a home owner. If the did, then they’d be a home owner.
