  We consider an equivariant analogue of a conjecture of Borcherds. Let $Y$ be
a real $K3$ surface without real points. Let $g$ be a Ricci-flat Kaehler metric
on $Y$ invariant under the complex conjugation. We shall prove that the
equivariant determinant of the Laplacian of $(Y,g)$ with respect to the complex
conjugation is expressed as the norm of the Borcherds Phi-function at the
"period point". Here the period is not the one in algebraic geometry.
