Take $N = \mathbf{Z}$, $N_\mathbf{R} = \mathbf{R}$. Then I know that the only cones are the intervals $\sigma_+ = [0,\infty)$, $\sigma_- =(-\infty,0]$, and $\tau = \{0\}$.
Consider the fans $\{\sigma_+,\tau\}$ and $\{\sigma_- , \tau\}$. The claim is that these both give the toric variety $\mathbf{C}$.
Let's focus on $\{\sigma_+,\tau\}$ and call $\sigma = \sigma_+$. This gives the affine toric variety
$$U_\sigma = \text{Spec } \mathbf{C} [\sigma ^ \vee \cap M ]$$
But $\sigma ^ \vee = \{0\}$! Therefore, $U_\sigma = \text{Spec } \mathbf{C} [\{0\}] \cong \text{Spec }\mathbf{C} [x] = \mathbf{C}$
Is this reasoning correct?