Hybrid particle–field molecular dynamics is a molecular simulation strategy, wherein particles couple to a density field instead of through ordinary pair potentials. Traditionally considered a mean-field theory, a momentum and energy-conserving hybrid particle–field formalism has recently been introduced, which was demonstrated to approach the Gaussian Core model potential in the grid-converged limit. Here, we expand on and generalize the correspondence between the Hamiltonian hybrid particle–field method and particle–particle pair potentials. Using the spectral procedure suggested by Bore and Cascella, we establish compatibility to any local soft pair potential in the limit of infinitesimal grid spacing. Furthermore, we document how the mean-field regime often observed in hybrid particle–field simulations is due to the systems under consideration, and not an inherent property of the model. Considering the Gaussian filter form, in particular, we demonstrate the ability of the Hamiltonian hybrid particle–field model to recover all structural and dynamical properties of the Gaussian Core model, including solid phases, a first-order phase transition, and anomalous transport properties. We quantify the impact of the grid spacing on the correspondence, as well as the effect of the particle–field filtering length scale on the emergent particle–particle correlations.