would it be 'mi casa su casa' or 'mi casa es su casa' or would you use tu?
You would have to use "su" not "tu" because if you use "tu" you are saying "My house is you house." Su means your, whereas Tu means you. It would be "Mi casa es su casa".
tu means your, tú means you, su means his/hers/its, so if you are trying to say "my house is your house.", you would say "mi casa es tu casa"
Or su can also mean you (formal), right? So that's probably why the saying is "mi casa (es) su casa."
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