  We give an expression for the Smith-Thom deficiency of the Hilbert square
$X^{[2]}$ of a smooth real algebraic variety $X$ in terms of the rank of a
suitable Mayer-Vietoris mapping in several situations. As a consequence, we
establish a necessary and sufficient condition for the maximality of $X^{[2]}$
in the case of projective complete intersections, and show that with a few
exceptions no real nonsingular projective complete intersection of even
dimension has maximal Hilbert square. We also provide new examples of smooth
real algebraic varieties with maximal Hilbert square.
