We propose a conceptually new class of dynamical experiments whose goal is to falsify the hypothesis that an interaction between quantum systems is mediated by a purely local classical field. The systems we study implement a dynamics that cannot be simulated by means of local operations and classical communication (LOCC), even when no entanglement is ever generated at any point in the process. Using tools from quantum information theory, we estimate the maximal fidelity of simulation that a local classical interaction could attain while employing only LOCC. Under our assumptions, if an experiment detects a fidelity larger than that calculated threshold, then a local classical description of the interaction is no longer possible. As a prominent application of this scheme, we study a general system of quantum harmonic oscillators initialised in normally distributed coherent states and interacting via Newtonian gravity, and discuss a possible physical implementation with torsion pendula. One of our main technical contributions is the calculation of the above bound on the maximal LOCC simulation fidelity for this family of systems. As opposed to existing tests based on the detection of gravitationally mediated entanglement, our proposal works with coherent states alone, and thus it does not require the generation of largely delocalised states of motion nor the detection of entanglement.