  Classification results for complex Riemannian foliations are obtained. For
open subsets of irreducible Hermitian symmetric spaces of compact type, where
one has explicit control over the curvature tensor, we completely classify such
foliations by studying the infinitesimal model associated to the canonical
connection. We also establish results for symmetric spaces of non-compact type
and a general rigidity result for any irreducible K\"ahler manifold.
