By this https://mathoverflow.net/q/126329 the moduli space of complex $d$-tori is $X = GL_{d}(\mathbb{C})\backslash GL_{2d}(\mathbb{R})/GL_{2d}(\mathbb{Z})$. What would happen if complex structures which are anti-biholomorphic equivalent were also identified? If the resulting space is $Y$ would the canonical map $Y \to X$ be a $2$-sheeted covering? Is there a reference for such moduli spaces?

I don't think so because if $d = 1$ it would be
$$X \cong \mathbb{R}^*SO(2)\backslash GL_2(\mathbb{R})/GL_2(\mathbb{Z}) \cong SO(2)\backslash SL_2(\mathbb{R})/SL_2(\mathbb{Z}) \cong \mathbb{H}/SL_2(\mathbb{Z})\cong \mathbb{R^2}$$ and $$Y \cong \mathbb{R}^*O(2)\backslash GL_2(\mathbb{R})/GL_2(\mathbb{Z}) \cong (SO(2)\backslash SL_2(\mathbb{R}))/GL_2(\mathbb{Z})\cong \mathbb{H}/GL_2(\mathbb{Z}) \cong \mathbb{R}^2_+ = \{(x,y)\in \mathbb{R}^2 \ | \ y \geq 0\}$$ so $X \to Y$ cannot be a $2$-sheeted covering?