  We describe a criterion for a real or complex hyperbolic lattice to admit a
RFRS tower that consists entirely of congruence subgroups. We use this to show
that certain Bianchi groups $\mathrm{PSL}(\mathcal{O}_d)$ are virtually fibered
on congruence subgroups, and also exhibit the first examples of RFRS K\"ahler
groups that are not a subgroup of a product of surface groups and abelian
groups.
