Let $K$ be a valued field, and consider the ring $R=K((x_1,\dots,x_m))$ of formal Laurent series. This is "the germ of the torus at $0$". Is there a theory of "local toric varieties" where $R$ replaces the usual ring $K[x_1^{\pm 1},\dots,x_m^{\pm 1}]$?
A remark: one easily sees that a necessary condition for a fan for define a local toric variety, is that the support of the fan lies in $(\mathbb{R}_{\geq 0})^m$. This makes sense, since none of the coordinate functions can have poles near $0$.
A next step would then be to tropicalize a subset $Y$ of $\text{Spec}(R)$, which should give a tropical variety $Trop(Y)$ inside $(\mathbb{R}_{\geq 0})^m$. When $K$ is given the trivial valuation, $Trop(Y)$ should be a fan, and any local toric variety whose fan is supported on $Trop(Y)$, should give a "compactification" of $Y$.
It is not clear to me how this should work. Is there any literature on this subject?
