We describe an algorithm which, on input a genus 2 curve C/ℚ whose Jacobian J/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/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.