By definition $X_{\sigma}^{{\text{\rm an}}}\subset\rho_{\Sigma}^{-1}(X_{\sigma}(\mathbb{R}_{\geq 0}))$. For the reverse inclusion we will write only the non-Archimedean case. Assume that $p\in\rho_{\Sigma}^{-1}(X_{\sigma}(\mathbb{R}_{\geq 0}))$. There is a $\sigma^{\prime}$ with $p\in X_{\sigma^{\prime}}^{{\text{\rm an}}}$. Let $\tau=\sigma\cap\sigma^{\prime}$ be the common face. Then $p$ is a multiplicative seminorm of $K[M_{\sigma^{\prime}}]$ and we show next that it can be extended to a multiplicative seminorm of $K[M_{\tau}]$. By [Ful93, §1.2 Proposition 2] there is an element $u\in M_{\sigma^{\prime}}$ such that $M_{\tau}=M_{\sigma^{\prime}}+\mathbb{Z}_{\geq 0}(-u)$. Hence $K[M_{\tau}]=K[M_{\sigma^{\prime}}+\mathbb{Z}_{\geq 0}(-u)]$. Since $\rho_{\Sigma}(p)\in X_{\tau}(\mathbb{R}_{\geq 0})$ we have that $|\chi^{u}(p)|\not=0$. Therefore $p$ extends to a multiplicative seminorm of $K[M_{\tau}]$. Hence $p\in X^{{\text{\rm an}}}_{\tau}\subset X^{{\text{\rm an}}}_{\sigma}$. ∎