Joint possessive with a personal pronoun

Karen asked me over chat how to phrase an invitation to her and her roommate Julie's apartment: "Come to [Julie and Karen's] apartment." After pondering for a few moments, I realized that it's a tricky question. If the person being addressed knew that Julie and Karen both shared ownership of the apartment and Julie had already been referenced, then "Come to our apartment" is an easy choice. If that's not the case, I believe the correct grammatical phrasing is "Come to Julie and my apartment." That sounds awkward. Referring to yourself in the third person ("Come to Julie and Karen's apartment") only works if you are Charles Barkley or just plain crazy.

I turned to Garner's Dictionary of Modern American Usage but didn't find this issue addressed under the heading Possessive(s) (perhaps it's covered in the newer edition). Apparently this is one of those blemishes of English that many usage guys brush under the rug, as I found an article on just this case on the web.

No easy answers. Faced with this linguistic tongue twister, no one will blame you for flipping the order of the possessors and writing, "Come to me and Julie's apartment."