  The ferromagnetic XY model on sparse random graphs in a randomly oriented
field is analyzed via the belief propagation algorithm. At variance with the
fully connected case and with the random field Ising model on the same
topology, we find strong evidences of a tiny region with Replica Symmetry
Breaking (RSB) in the limit of very low temperatures. This RSB phase is robust
against different choices of the external field direction, while it rapidly
vanishes when increasing the graph mean degree, the temperature or the
directional bias in the external field. The crucial ingredients to have such a
RSB phase seem to be the continuous nature of vector spins, mostly preserved by
the O(2)-invariant random field, and the strong spatial heterogeneity, due to
graph sparsity. We also uncover that the ferromagnetic phase can be marginally
stable despite the presence of the random field. Finally, we study the proper
correlation functions approaching the critical points to identify the ones that
become more critical.
