A construction of algebraizable formal models

Let $X$ be a variety over a complete nontrivially valued field $K$. We construct an algebraizable formal model for the analytification of $X$ in the case $X$ admits a closed embedding into a toric variety. By algebraizable we mean that the formal model is given by the completion along the special fiber of a locally finite type flat scheme over the valuation ring $K^\circ$. We construct the formal model via the combinatorial theory of $\mathbb{T}$-toric varieties over $K^{\circ}$.