Let $A$ be a domain, and $K$ its field of fractions. The group $K^*/A^*$ is partially ordered by the divisibility relation $x | y$ if $yA \subseteq xA$.
Is there some characterization of all partially ordered Abelian groups $G$ that can occur in this way?
I see (writing $G$ additively) that any element of $G$ must be of the form $x - y$, where $x, y \geq 0$, but I don't imagine that that will be sufficient.
I wanted to post this as a comment but it is too long.
Here is another remark: if moreover $A$ is a PID, then every group occurring this way is a lattice-ordered group, that is, a group such that the underlying PO set is a lattice. The theory is the subject of the book "Theory of lattice-ordered groups" by Darner.
Here is a proof sketch. By proposition 3.3 in Darner, it suffices to check that every two elements have a least upper bound. So given $x, y \in K^*$, we need to show that there exists $z \in K^*$ such that $zA = xA \cap yA$. This follows from the fact that a submodule of a cyclic module over a PID is cyclic (a corollary of the structure theorem).
You should take a look at chapter 9 of Darner, which deals with abelian lattice-ordered groups, for which many strong results are available. For instance, every abelian lattice-ordered group is representable, that is, isomorphic to a subdirect product of totally ordered groups. Maybe there is a way to combine this with Captain Lama's comment to get a characterization for the case in which $A$ is a PID.