  Let $\mathcal{X}$ be a complex projective manifold of dimension $n$ defined
over the reals and let $M$ be its real locus. We study the vanishing locus
$Z\_{s\_d}$ in $M$ of a random real holomorphic section $s\_d$ of $\mathcal{E}
\otimes \mathcal{L}^d$, where $\mathcal{L} \to \mathcal{X}$ is an ample line
bundle and $\mathcal{E} \to \mathcal{X}$ is a rank $r$ Hermitian bundle, $r \in
\{1,\dots, n\}$. We establish the asymptotic of the variance of the linear
statistics associated with $Z\_{s\_d}$, as $d$ goes to infinity. This
asymptotic is of order $d^{r-\frac{n}{2}}$. As a special case, we get the
asymptotic variance of the volume of $Z\_{s\_d}$. The present paper extends the
results of [20], by the first-named author, in essentially two ways. First, our
main theorem covers the case of maximal codimension ($r = n$), which was left
out in [20]. And second, we show that the leading constant in our asymptotic is
positive. This last result is proved by studying the Wiener--It{\=o} expansion
of the linear statistics associated with the common zero set in $\mathbb{RP}^n$
of $r$ independent Kostlan--Shub--Smale polynomials.
