Despite its fully unitary dynamics, the bosonic Kitaev chain (BKC) displays key hallmarks of non-Hermitian physics, including nonreciprocal transport and the non-Hermitian skin effect. Here, we demonstrate another remarkable phenomena: the existence of an entanglement phase transition (EPT) in a variant of the BKC that occurs as a function of a Hamiltonian parameter g and which coincides with a transition from a reciprocal to a nonreciprocal phase. As g is reduced below a critical value, the postquench entanglement entropy of a subsystem of size l goes from a volume-law phase, where it scales as l, to a supervolume-law phase, where it scales like lN, where N is the total system size. This EPT occurs for a system undergoing purely unitary evolution and does not involve measurements, postselection, disorder, or dissipation. We derive analytically the entanglement entropy out of and at the critical point for the cases of l=1 and l/N1.