Control of thermal transport at the nanoscale is of great current interest for creating novel thermal logic and energy conversion devices. Recent experimental studies have demonstrated that radiative heat transfer between macroscopic objects separated by nanogaps, or between nanostructures located in the far-field of each other, can exceed the blackbody limit. Here, we show that the radiative heat transfer between two coplanar SiN membranes can be modulated by factors as large as five by bringing a third planar object into close proximity of the membranes. Numerical modelling reveals that this modulation is due to a modification of guided modes (supported in the SiN nanomembranes) by evanescent interactions with the third object. This multi-body effect could offer an efficient pathway for active control of heat currents at the nanoscale.