  Fix an integer $d>0$. In 2008, David and Weston showed that, on average, an
elliptic curve over $\mathbf{Q}$ picks up a nontrivial $p$-torsion point
defined over a finite extension $K$ of the $p$-adics of degree at most $d$ for
only finitely many primes $p$. This paper proves an analogous averaging result
for principally polarized abelian surfaces over $\mathbf{Q}$ with real
multiplication by $\mathbf{Q}(\sqrt{5})$ and a level-$\sqrt{5}$ structure.
Furthermore, we indicate how the result on abelian surfaces with real
multiplication by $\mathbf{Q}(\sqrt{5})$ relates to the deformation theory of
modular Galois representations.
