Congruence RFRS towers

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 PSL(Od) are virtually fibered on congruence subgroups, and also exhibit the first examples of RFRS Kähler groups that are not a subgroup of a product of surface groups and abelian groups.