$\newcommand{\complex}{\mathbb{C}}\newcommand{\real}{\mathbb{R}}\newcommand{\proj}{\mathbb{P}}\newcommand{\Hom}{\text{Hom}}$I apologize in advance for my naïve understanding of real algebraic geometry. I define a real projective variety to be a subset of $\mathbb{P}(\real^n)$ that is the zero locus of some finite collection of real homogeneous polynomials $f_1,...,f_p \in \real[x_1,...,x_n]$.

Let $X, Y \subseteq \mathbb{P}(\real^n)$ be real projective varieties, and let

$$Seg: \proj(\real^n)\times \proj(\real^n) \rightarrow \proj(\real^{n}\otimes \real^n)$$

be the Segre embedding $(v,w)\mapsto v\otimes w$. Is it true that $Seg(X \times Y) \subseteq \proj(\real^n \otimes \real^n)$ is a real projective variety?

Assuming that $Seg(X \times Y)$ is a real projective variety, I have a follow-up question: Let $\Pi\in \Hom_{\real}(\real^n \otimes \real^n)$ be the linear map that acts as $\Pi(v \otimes w)=\frac{1}{2}(v\otimes w + w \otimes v)$. Is it true that $\Pi(Seg(X \times Y)) \subseteq \proj(\real^n \otimes \real^n)$ is a real projective variety?