A basic multiset identity says:
$$A+B = (A \cap B) + (A \cup B)$$
Allowing ourselves to use negative multiplicities and rearranging:
$$A-(A \cap B) = (A \cup B)-B$$
But since $A \supseteq (A \cap B),$ hence the LHS denotes a genuine (non-negative) multiset. Equivalently, since $(A \cup B) \supseteq B$, hence the RHS denotes a multiset. Of course, they're equal, so we've really just verified the same thing twice!
Anyway define that $A \mathop{-\!-} B$ equals $A - (A \cap B)$, or else $(A \cup B)-B$ if you prefer.
Question. Has anyone ever suggested a name or notation for the operation $\mathop{-\!-}$ on multisets?
This shows up naturally in the following way. Suppose we're living in a circle of $n$ beads. We begin taking steps of size $m$. How many steps must we take before we return to the starting point? Easy! Write $[n]$ for the multiset of prime factors of $n$ and $[m]$ for the multiset of prime factors of $n$. Write $\Pi(M)$ for the product of the elements of any multiset $M$ whose elements are numbers. Then the number of steps before we return to our starting point is $$\Pi([n]\mathop{-\!-}\,[m]).$$
(I haven't proved this, so its potentially completely wrong...)
In naive set theory, we usually have the operation $$A \setminus B := \{x: x \in A, x \notin B\}$$ For ordinary sets, we have: $$A \setminus B = A - (A \cap B) = (A \cup B) - B$$ although usually the latter two are also written using $\setminus$.
Therefore, it would seem reasonable to retain this definition for multisets. Indeed, this definition also preserves the identities with $\setminus$ for multisets: $$A \setminus B = A \setminus (A \cap B) = (A \cup B) \setminus B$$