Realification determinant identity (${\det }_{\mathbb{R}}\mathcal{R}(A)=|{\det }_{\mathbb{C}}A{|}^2$)

The realification of a complex matrix has real determinant $|{\det }_{\mathbb{C}}A{|}^2$. Indeed the invertible complex coordinate change $(x,y)\mapsto (x+iy,x-iy)$ conjugates its complexified block matrix to $\mathrm{diag}(A,\overline{A})$. Taking determinants gives the identity, including singular matrices.