“Paging Mr. Sarsaparilla, Mr. Rufus Xavier Sarsaparilla, line one, white courtesy phone.”
Christ, we’ll post it again until somebody reads it…
From EnderX’s post: "When one of the possessors in a compound possessive is a personal pronoun, we have to put both possessors in the possessive form or we end up with something silly: “Bill and my car had to be towed last night.”
It is not enough to be grammatically correct: the best option is a form with balances lack of ambiguity with brevity. If expressing ownership is important then it should be expressed as explicitly as possible; if you don’t mind if people believe you and your sister might own them separately, or if you can trust them to already know you have cats in common, it becomes simpler.
I am not sure that “me and my sister’s cats” would be unambiguously interpreted as meaning that the cat is shared, and “my sister’s and my cats” (if correct) is ambiguous, as is “the cats of me and my sister”. If it is important to establish co-ownership, then say something like:
“The cats me and my sister share”
“The cats shared by me and my sister”
If ownership isn’t important, it becomes far simpler, and you can use already-mentioned forms: e.g. “the cats of my sister and me”, which is grammatical but ambiguous.
Duh! “The cats my sister and I share” or “The cats shared by my sister and I”. There’s an unbendable natural law about grammatical posts.
The first example is correct nominative case; the second example needs objective case since the pronoun is part of the object of the preposition. Thus, “The cats shared by my sister and me.”
Ex-English teacher here. Sorry about that.
I’m with you on the second, but should the first one not be “The cats I and my sister share” (or, even better, “The cats my sister and I share”)?
Now why did my post take that long to appear?
You’re absolutely right; sorry I missed it. The correct form is “Bill’s and my car…”
Well no, because in my example “sisters” is supposed to refer to my sister and I (humans). 
Julie
Ah, but cats believe that their owners are their pets…
The apostrophe is being added to the entire noun phrase. “My” is only modifying “sister.”
{ [My sister] and me } 's cats
curly chick, jjimm, rk, I am not joking and I’m perfectly correct. I do notice however that I’m in Texas and you’re over in England and Ireland.
While the alternates you present sound fine to me in a grammatical sense, I just wouldn’t ever say it that way. It sounds stilted or officious to me. The phrasing I offered is the most natural-sounding choice to my ears.
Sticking the possessive at the end of a noun phrase isn’t remarkable in any way, we do it all the time: “The man with the red car’s shopping cart is broken” or even “The girl who came with me’s car is over there.” The only other consideration is which pronoun to use, “me” or “I.” In English we use the oblique form with conjunctions, so it’s “me.”
I prefer “my sister and me’s cats” to “me and my sister’s cats” because the latter is potentially ambiguous.
-fh
“Me’s” isn’t a word, hazel-rah. The first-person possessive is “mine.”
I see hazel-rah’s point (adding the apostrophe-s to the end of what is essentially a descriptive phrase (“the man with the red car” or “the girl who came with me”)). But while this is generally understandable in colloquial spoken English, it’s not technically grammatically correct, and it’s kind of messy when written down. It’s one of those things your English teacher would label “Awkward - rephrase”.
That being said, “and me” is not a descriptive phrase belonging to “my sister”, so even the colloquial rule doesn’t apply. The double possessive is required to avoid ambiguity. Note:
Me and my sister’s cat: refers to me and the cat, not just the cat.
My sister and my cat: same problem.
My sister’s and my cat: the phrase “and my” temporarily defers the object of the possessive “my sister’s”, in the same way you might refer to “medium- and long-term plans”; the word “term”, like the word “cat”, is implied in the first case.
Wow, am I a pedant.
Given the unfortunate modern usage of apostrophes denoting plural, you’re still going to cause some confusion among the subliterate who think you’re talking about your multiple sisters and your cat.
Damned if you do and damned if you don’t.
Whoops, I meant “my.”
“Me’s” is no more a word than “I’s” is. That is to say, it ain’t.
The OP asked if it was possible to say this correctly, not write it correctly. In American English “my sister and me’s cat” is correct. Technically, grammatically, any way you want to slice it. It is correct.
I don’t know what you mean by colloquial rule. Or, for that matter, descriptive phrase. Could you clarify?
dantheman, I know what the first-person possessive is. Try and pay attention: the possessive 's in “my sister and me’s cat” is modifying the entire noun phrase.
-fh
hazel-rah - “me’s” is not a word. Please stop saying it is. You cannot, in the rules of English, add an apostrophe-s to the word “me.” To state otherwise is to propagate a falsehood.
And an apostrophe-s doesn’t “modify” anything. It changes a regular noun into a possessive noun. It’s not an adjective or adverb, words that do modify other words.
Finally, you can’t add apostrophe-s to an entire phrase.
hazel’s answer seems right to me. I know where the objectors are coming from: it looks wrong to see the form “me’s” in writing, but no good alternative presents itself.
dantheman’s style book recommends a form which it admits is inconsistent, but has the advantage of not sounding “silly”. However, this is at the cost of ambiguity.
If we can in general distinguish between shared and separate ownership relations by using the possessive form of one or both of the owners, then it seems wrong-headed to lose that distinction in this case.
Huh? How could it possibly be correct? There’s no such word.
What was inconsistent about the answer I gave? It’s completely accurate without being awkward.