$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SL{SL}$By Will Sawin's answer to Moduli Spaces of Higher Dimensional Complex Tori 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?