Let $N=\mathbb{Z}^{2}$ and consider the fan $\Sigma$ generated by $e_{0}=(-1,-1)$, $e_{1}=(1,0)$ and $e_{2}=(0,1)$. Then $X_{\Sigma}=\mathbb{P}^{2}$. The virtual support function $\Psi=0$ corresponds to the trivial line bundle $\mathcal{O}_{\mathbb{P}^{2}}$. Consider the function

$$\psi(x,y)=\begin{cases}0,&\text{ if }x\leq 0,\\ x,&\text{ if }0\leq x\leq 1,\\ 1,&\text{ if }1\leq x.\\ \end{cases}$$

Then $\operatorname{rec}(\psi)=\Psi$, but $\psi$ does not extend to a continuous function on $N_{\Sigma}$ and therefore it does not determine a model of $(X_{\Sigma},\mathcal{O})$. By contrast, let $\Sigma^{\prime}$ be the fan obtained subdividing $\Sigma$ by adding the edge corresponding to $e^{\prime}=(0,-1)$. Then $X_{\Sigma^{\prime}}$ is isomorphic to a blow-up of $\mathbb{P}^{2}$ at one point. The function $\psi$ extends to a continuous function on $N_{\Sigma^{\prime}}$ and it corresponds to a toric model of $(X_{\Sigma^{\prime}},\mathcal{O})$.