If you view a $\mathbb{Z}$-valued Cartier divisor (on say, an integral separated scheme X) as a $\mathbb{G}\_m$-torsor on X with generic trivialization, then for any torus T with character group $X^\ast(T)$, an $X^\ast(T)$-valued divisor on X is a T-torsor with generic trivialization.  The analogous construction will work for any group of multiplicative type, if you replace the character group with a suitable fpqc sheaf of abelian groups on the base.  This shows up for X a curve in some treatments of geometric class field theory, since the space of these divisors is the affine Grassmannian (in the sense of Beilinson-Drinfeld) for T over the Ran space of X.  This space is used in, e.g., Gaitsgory's Twisted Whittaker model paper, where it forms a home (together with some similar objects) for factorizable sheaves.

If R is a number ring, you can demand that $X^\ast(T)$ be a sheaf of R-modules, so in this case, you're looking for torsors under tori with CM, with generic trivializations.  I don't know where or if this is used, but it seems interesting enough.