*IN GENERAL, the way to make joint possession for compound nouns is to make only the second noun possessive by adding ‘s to the end. (Example: Bill and Ted’s excellent house is blue.)
But when one of the possessors is a personal pronoun, it doesn’t make sense to follow that rule. We have to put BOTH possessors in the possessive form or we end up with something silly. (ex: Napoleon and my house is white = incorrect; It should be Napoleon’s and my house is white.)
How about when “I” is involved? For example, the sentence “That is Medusa and I’s snake” is incorrect for two reasons.
1) The writer did follow the general rule of thumb by making the second noun possessive. However, since one is a pronoun, BOTH have to be in a possessive form for clarity. So, in this case, “Medusa” becomes “Medusa’s.”
2) The “I” in this sentence is a subject pronoun. It needs to be in the possessive form, which is “my.” Therefore, the sentence, “That is Medusa’s and I’s snake” would be correctly rewritten to say, “That is Medusa’s and my snake.”
The construction may sound awkward to you, but it is correct. However, if you don’t like it, you can always avoid it entirely and go the simple route of saying OUR, as in, “That is OUR snake.”
Another way to rewrite it is, “That snake belongs to Medusa and ME.” (It’s ME here in this sentence, not I or MY, because ME is the object pronoun in this sentence, not a subject or possessive pronoun.)