Corresponding to a strongly convex rational polyhedral cone $\sigma$ in the lattice $N$ we have the affine toric variety $U_\sigma=\operatorname{Hom_{semi group}}(\sigma^\vee\cap M,\mathbb C)$ where $M$ is the dual lattice.
Fulton, defines the real part of this variety as $(U_\sigma)_\mathbb R=\operatorname{Hom_{semi group}}(\sigma^\vee\cap M,\mathbb R)$.
Now if $N=\mathbb Z^2$ with $e_1,e_2$ as the standard basis and $\sigma=\mathbb R_{\ge0}e_1+\mathbb R_{\ge0}e_2$ the $U_\sigma$ can be identified with $\mathbb C^2$ by the map $$(\chi^{e_1},\chi^{e_2}):U_\sigma\to\mathbb C^2\quad\text{ which sends}\quad u\mapsto(u(e_1),u(e_2))$$
This basically comes from identifying $\mathbb C[x,y]$ with $\mathbb C[\chi^{e_1},\chi^{e_2}]$ and taking $\operatorname{Maxspec}$. However one can't do the same thing with $(U_\sigma)_\mathbb R$ as $\mathbb R$ is not algebraically closed. So in that case,
What would $(U_\sigma)_\mathbb R$ look like?
Thank you.