Kooyeen,
If you want to know how that idea is expressed in real spoken American English, here's how:
They are almost always reworded to avoid a possessive adjective (
my,
your, etc.) joined to another noun by
and. Because of this, the 'problem' you pose is almost entirely theoretical.
Bill and my dog becomes
the dog that belongs to Bill and me, for example, or
the dog that Bill and I have/own.
Normally the conversation has already mentioned
Bill and I/me in the context, so, as suggested above,
our dog becomes the solution.
You may be surprised to hear this, but when a speaker, not having
planned ahead, has painted himself into a grammatical corner on this
concept, it is not unusual for the choice to be
Bill and I's dog.
CJ