  We classify all $\Sp_4(\mathbb{C})$-rigid, quasi-unipotent local systems and
show that all of them have geometric origin. Furthermore, we investigate which
of those having a maximal unipotent element are induced by fourth order
Calabi-Yau operators. Via this approach, we reconstruct all known Calabi-Yau
operators inducing a $\Sp_4(\mathbb{C})$-rigid monodromy tuple and obtain
closed formulae for special solutions of them.
