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.  This shows up for X a curve, e.g., in some treatments of geometric class field theory and Gaitsgory's Twisted Whittaker model paper.  The analogous construction will work for any group of multiplicative type, replacing the character group with a suitable fpqc sheaf of abelian groups on the base.

If R is a number ring, you can demand that $X^\ast(T)$ be an R-module, 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.