  Let $\mathcal{O}$ be a valuation ring of height one of residual
characteristic exponent $p$ and with algebraically closed field of fractions.
Our main result provides a best possible resolution of the monoidal structure
$M_X$ of a log variety $X$ over $\calO$ with a vertical log structure: there
exists a log modification $Y\to X$ such that the monoidal structure of $Y$ is
polystable. In particular, if $X$ is log smooth over $\mathcal{O}$, then $Y$ is
polystable with a smooth generic fiber. As a corollary we deduce that any
variety over $\mathcal{O}$ possesses a polystable alteration of degreee $p^n$.
The core of our proof is a subdivision result for polyhedral complexes
satisfying certain rationality conditions.
