  We continue our study of integral points on moduli schemes by combining the
method of Faltings (Arakelov, Parsin, Szpiro) with modularity results and
Masser-W\"ustholz isogeny estimates. In this work we explicitly bound the
height and the number of integral points on coarse Hilbert moduli schemes
outside the branch locus. In the first part we define and study coarse Hilbert
moduli schemes with their heights and branch loci. In the second part we
establish the effective Shafarevich conjecture for abelian varieties $A$ over a
number field $K$ such that $A_{\bar{K}}$ has CM or $A_{\bar{K}}$ is of GL2-type
and isogenous to all its $G_\mathbb Q$-conjugates. In the third part we
continue our explicit study of the Parsin construction given by the forgetful
morphism of Hilbert moduli schemes. We now work out our strategy for arbitrary
number fields $K$ and we explicitly bound the number of polarizations and
module structures on abelian varieties over $K$ with real multiplications. In
the last part we illustrate our results by applying them to two classical
surfaces first studied by Clebsch (1871) and Klein (1873): We explicitly bound
the Weil height and the number of their integral points.
