  We introduce a quantum Monte Carlo algorithm to measure the R\'enyi
entanglement entropies in systems of interacting bosons in the continuum. This
approach is based on a path integral ground state method that can be applied to
interacting itinerant bosons in any spatial dimension with direct relevance to
experimental systems of quantum fluids. We demonstrate how it may be used to
compute spatial mode entanglement, particle partitioned entanglement, and the
entanglement of particles, providing insights into quantum correlations
generated by fluctuations, indistinguishability and interactions. We present
proof-of-principle calculations, and benchmark against an exactly soluble model
of interacting bosons in one spatial dimension. As this algorithm retains the
fundamental polynomial scaling of quantum Monte Carlo when applied to
sign-problem-free models, future applications should allow for the study of
entanglement entropy in large scale many-body systems of interacting bosons.
