  We describe an algorithm which, on input a genus $2$ curve $C/\mathbb{Q}$
whose Jacobian $J/\mathbb{Q}$ has real multiplication by a quadratic order in
which $7$ splits, outputs twists of the Klein quartic curve parametrising
elliptic curves whose mod $7$ Galois representations are isomorphic to a
sub-representation of the mod $7$ Galois representation attached to
$J/\mathbb{Q}$. Applying this algorithm to genus $2$ curves of small conductor
in families of Bending and Elkies--Kumar we exhibit a number of genus $2$
Jacobians whose Tate--Shafarevich groups (unconditionally) contain a
non-trivial element of order $7$ which is visible in an abelian three-fold.
