  We generalise the notion of a separating intersection of links (SIL) to give
necessary and sufficient criteria on the defining graph $\Gamma$ of a
right-angled Coxeter group $W_\Gamma$ so that its outer automorphism group is
large: that is, it contains a finite index subgroup that admits the free group
$F_2$ as a quotient. When $Out(W_\Gamma)$ is not large, we show it is virtually
abelian. We also show that the same dichotomy holds for the outer automorphism
groups of graph products of finite abelian groups. As a consequence, these
groups have property (T) if and only if they are finite, or equivalently
$\Gamma$ contains no SIL.
