  The geodesic complexity of a Riemannian manifold is a numerical isometry
invariant that is determined by the structure of its cut loci. In this article
we study decompositions of cut loci over whose components the tangent cut loci
fiber in a convenient way. We establish a new upper bound for geodesic
complexity in terms of such decompositions. As an application, we obtain
estimates for the geodesic complexity of certain classes of homogeneous
manifolds. In particular, we compute the geodesic complexity of complex and
quaternionic projective spaces with their standard symmetric metrics.
