Surface phonon polaritons (SPhPs) are coupled photon-phonon excitations that emerge at the surfaces of nanostructured materials. Although they strongly influence the optical and thermal behavior of nanomaterials, no technique has been able to reveal the complete three-dimensional (3D) vectorial picture of their electromagnetic density of states. Using a highly monochromated electron beam in a scanning transmission electron microscope, we could visualize varying SPhP signatures from nanoscale MgO cubes as a function of the beam position, energy loss, and tilt angle. The SPhPs' response was described in terms of eigenmodes and used to tomographically reconstruct the phononic surface electromagnetic fields of the object. Such 3D information promises insights in nanoscale physical phenomena and is invaluable to the design and optimization of nanostructures for fascinating new uses.