Let $\mathfrak{D}$ be such a subdivision. We will construct a canonical formal scheme $\mathfrak{X}^{\prime\prime}$ over $K^{\circ}$ associated to $\mathfrak{D}$ together with a morphism $\iota:\mathfrak{X}^{\prime\prime}\rightarrow\mathfrak{X}$ which induces the identity on the generic fibre such that there is a one to one correspondence between the open faces of $\mathfrak{D}$ and the strata of $\tilde{\mathfrak{X}}^{\prime\prime}$. First of all we choose a covering of $\mathfrak{X}$ as in Proposition 2.5. Let $\mathfrak{U}$ be a member of this covering with an étale morphism $\psi:\mathfrak{U}\rightarrow\mathfrak{X}(\boldsymbol{n},\boldsymbol{a},m)$ and let $S$ be the distinguished stratum of $\mathfrak{U}$. For $\Delta\in\mathfrak{D}\cap\Delta_{S}$ we set

$$A^{\prime}:=\left\{\sum_{\mu}a_{\mu}T^{\mu}\in K((T_{00},...,T_{p,n_{p}}))\;\Big|\;\forall_{u\in\Delta}:\;\lim v(a_{\mu})+u\cdot\mu=\infty\right\}$$

and $A:=A^{\prime}/(T_{00}...T_{0,n_{0}}-a_{0},...,T_{p0}...T_{p,n_{p}}-a_{p})$ and define

$$A^{\Delta}:=\left\{\sum_{\mu}a_{\mu}T^{\mu}\in A\;\Big|\;\forall_{u\in\Delta,\mu\in\mathbb{Z}^{\boldsymbol{n}+\boldsymbol{1}}}:\;v(a_{\mu})+\mu\cdot u\geq 0\right\}$$

and $\mathfrak{U}_{\Delta}:=\Spf A^{\Delta}$. If $\Delta_{1},\Delta_{2}\in\mathfrak{D}\cap\Delta_{S}$ then $\Delta_{1}\cap\Delta_{2}$ is a face of both and by transferring the arguments in [Gub13, Proposition 6.12] to the analytic situation, we obtain that the canonical morphisms $\mathfrak{U}_{\Delta_{1}\cap\Delta_{2}}\rightarrow\mathfrak{U}_{\Delta_{i}}$ are open immersions. Hence we can glue the $\mathfrak{U}_{\Delta}$ along this data to obtain a formal scheme which we denote by $\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}$ together with a morphism $\iota^{\prime}:\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}\rightarrow\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})$. Let $\psi^{\prime}:\mathfrak{U}^{\prime\prime}\rightarrow\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}\times\mathfrak{X}(m)$ be the base change of $\psi$ with respect to $\iota^{\prime}\times\Id$. The construction of $\mathfrak{U}^{\prime\prime}$ does not depend on the choice of $\psi$ up to isomorphism: Let $\rho:\mathfrak{U}\rightarrow\mathfrak{X}(\boldsymbol{n},\boldsymbol{a},m)$ be another étale morphism. Then up to reordering the coordinates, $\rho^{\ast}x_{i}=u_{i}\psi^{\ast}x_{i}$ for some $u_{i}\in\mathcal{O}(\mathfrak{U})^{\times}$. Then we have canonical $K^{\circ}$-algebra isomorphisms:

	$\displaystyle\mathcal{O}(\mathfrak{U})\hat{\otimes}_{\psi^{\ast}}A^{\Delta}$	$\displaystyle\rightarrow\mathcal{O}(\mathfrak{U})\hat{\otimes}_{\rho^{\ast}}A^{\Delta},$
	$\displaystyle a\otimes x_{i}$	$\displaystyle\mapsto u_{i}a\otimes x_{i},$

which yield an isomorphism of the $\mathfrak{U}^{\prime\prime}$ constructed with $\psi$ respectively $\rho$.
We glue the $\mathfrak{U}^{\prime\prime}$ to obtain our formal scheme $\mathfrak{X}^{\prime\prime}$. Although $\mathfrak{X}^{\prime\prime}$ might not be admissible, we can define its generic fibre and reduction map in the usual way as the algebras $A^{\Delta}\otimes_{K^{\circ}}K$ are strictly $K$-affinoid (see [Gub13, Proposition 6.17]). Then $\iota$ induces the identity on the generic fibres and we set $p_{\mathfrak{X}^{\prime\prime}}:=p_{\mathfrak{X}}$. Note that $\mathfrak{X}^{\prime\prime}$ is admissible if the vertices of the polytopes in $\mathfrak{D}$ are $\Gamma$-rational, in particular the base change of $\mathfrak{X}^{\prime\prime}$ to the valuation ring of the completion of an algebraic closure of $K$ is admissible, see [Gub13, Proposition 6.7].