  Let $p$ be an odd prime. For a number field $K$, we let $K_\infty$ be the
maximal unramified pro-$p$ extension of $K$; we call the group
$\mathrm{Gal}(K_\infty/K)$ the $p$-class tower group of $K$. In a previous
work, as a non-abelian generalization of the work of Cohen and Lenstra on ideal
class groups, we studied how likely it is that a given finite $p$-group occurs
as the $p$-class tower group of an imaginary quadratic field. Here we do the
same for an arbitrary real quadratic field $K$ as base. As before, the action
of $\mathrm{Gal}(K/\mathbb{Q})$ on the $p$-class tower group of $K$ plays a
crucial role; however, the presence of units of infinite order in the ground
field significantly complicates the possibilities for groups that can occur. We
also sharpen our results in the imaginary quadratic field case by removing a
certain hypothesis, using ideas of Boston and Wood. In an appendix, we show how
the probabilities introduced for finite $p$-groups can be extended in a
consistent way to the infinite pro-$p$ groups which can arise in both the real
and imaginary quadratic settings.}
