  The geometric torsion conjecture asserts that the torsion part of the
Mordell--Weil group of a family of abelian varieties over a complex
quasiprojective curve is uniformly bounded in terms of the genus of the curve.
We prove the conjecture for abelian varieties with real multiplication,
uniformly in the field of multiplication. Fixing the field, we furthermore show
that the torsion is bounded in terms of the $\mathit{gonality}$ of the base
curve, which is the closer analog of the arithmetic conjecture. The proof is a
hybrid technique employing both the hyperbolic and algebraic geometry of the
toroidal compactifications of the Hilbert modular varieties $\overline X(1)$
parametrizing such abelian varieties. We show that only finitely many torsion
covers $\overline X_1(\mathfrak{n})$ contain $d$-gonal curves outside of the
boundary for any fixed $d$. We further show the same is true for entire curves
$\mathbb{C}\rightarrow \overline X_1(\mathfrak{n})$.
