The house purchase documents reflect an equal ownership share between the buyers - same as any other joint ownership situation that happens. We’re both on the deed, etc.

We both have wills, which deals with with the probate issue of our ownership shares. You should have a will, even if you’re married! Probate can be expensive for your successors and family, and the process without a will is much more involved with the state. I also recommend establishing a revocable trust for these reasons.

I think the idea of a legal structured process for other belongings in event of a break-up sounds appealing in some regards, but reviews from friends who have been through the process rate it at “0 out of 10 this SUCKS” and “now I owe alimony for the rest of my life-ish because my ex decided to quit their job and become a grad student before we split”. In any event if we break up acrimoniously, we can still hire lawyers…