Sometimes mutually beneficial walls like that are a product of an agreement, which if properly executed should be of record (where your deed is recorded). That may be a start (and possibly an answer too). Check your title documents too, maybe there is some explanation (although it will probably be just an exclusion on the policy).
They may also be in place by the developer, plat, restrictions, or addressed in HOA records.
Where is the wall located? On your property or theirs? Who built it (ie which previous owner)?
If it seems as if ownership belongs to the other, I don't think you can force repairs at this time. However if it has already shown signs of failure, you may hint that you will not tolerate the boundary line encroachment and maybe that will open negotiations toward repairs, before it is too late and you are forced to file a complaint for damages.
If the wall is on your property you may not have as much leverage in having them chip in to repair it. If it fails and he argues you had an opportunity to reduce damage (and he will), you might end up having to bear the brunt of the clean up and repairs to the wall.
All this is speculative, and depends on your jurisdiction. I'd see if you can consult with an attorney (on the sly) for an initial consultation, but bring answers to the above questions with you.
Good luck.
edit: Initial consultation with a Real Estate Attorney (or preferably a specialist such as a RE Title Attorney). No reason to consult with a litigation attorney just yet

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edit2: I also forgot to mention it could be a party wall, which typically is involved with PUDs or HOAs which title documents will have the answer you are looking for. If not, it is the same common property as the common areas in a condo or townhouse set up, where each member is partial owner, and partially responsible in relation to ownership rights (ie 1/2, 1/5, etc...)