Explicit description of the space of global sections of a torus invariant Weil divisor over a real toric variety

Question

Let $D$ be a Weil divisor on a normal toric variety $X$ with fan $\Sigma$ that is invariant under the action by the torus $T$. Then Proposition 4.3.2 of the textbook Toric Varieties by Cox, Little and Schenck says that the vector space of global sections of the invertible sheaf $\mathcal{O}_X(D)$ associated to $D$ is given by: $$\Gamma(X, \mathcal{O}_X(D)) \cong \bigoplus_{\operatorname{div}(\chi^m) + D \ge 0} \mathbb{C} \cdot \chi^m,$$ where $\operatorname{div}(\chi^m)$ is the divisor associated to a character $\chi^m$ of $T$. The toric varieties in their textbook are defined over the complex numbers.

Does an analogous theorem holds for real toric varieties? Namely, if $Y$ is a normal real toric variety and $D$ is a torus invariant Weil divisor on $Y$, is there an isomorphism: $$\Gamma(Y, \mathcal{O}_X(D)) \cong \bigoplus_{\operatorname{div}(\chi^m) + D \ge 0} \mathbb{R} \cdot \chi^m?$$

Answer 1

Yes, since the complex global sections are isomorphic to the tensor product of the real global sections with $\mathbb C$ by flat base change, and this isomorphism is equivalent for the action of the real torus, so when we write the complex global sections as a sum of eigenspaces this also writes the real global sections as a sum of eigenspaces.