  Let $\overline{\rho}: G_{\mathbf{Q}} \rightarrow {\rm GSp}_4(\mathbf{F}_3)$
be a continuous Galois representation with cyclotomic similitude character --
or, what turns out to be equivalent, the Galois representation associated to
the $3$-torsion of a principally polarized abelian surface $A/\mathbf{Q}$. We
prove that the moduli space $\mathcal{A}_2(\overline{\rho})$ of principally
polarized abelian surfaces $B/\mathbf{Q}$ admitting a symplectic isomorphism
$B[3] \simeq \overline{\rho}$ of Galois representations is never rational over
$\mathbf{Q}$ when $\overline{\rho}$ is surjective, even though it is both
rational over $\mathbf{C}$ and unirational over $\mathbf{Q}$ via a map of
degree $6$.
