Accessing the low-energy non-equilibrium dynamics of materials and their polaritons with simultaneous high spatial and temporal resolution has been a bold frontier of electron microscopy in recent years. One of the main challenges lies in the ability to retrieve extremely weak signals and simultaneously disentangling the amplitude and phase information. Here we present free-electron Ramsey imaging—a microscopy approach based on light-induced electron modulation that enables the coherent amplification of optical near fields in electron imaging. We provide simultaneous time-, space- and phase-resolved measurements of a micro-drum made from a hexagonal boron nitride membrane, visualizing the sub-cycle dynamics of two-dimensional polariton wavepackets therein. The phase-resolved measurement reveals vortex–anti-vortex singularities on the polariton wavefronts, together with an intriguing phenomenon of a travelling wave mimicking the amplitude profile of a standing wave. Our experiments show a 20-fold coherent amplification of the near-field signal compared with conventional electron near-field imaging, resolving peak field intensities in the order of a few watts per square centimetre, corresponding to field amplitudes of a few kilovolts per metre. As a result, our work paves the way for the spatiotemporal electron microscopy of biological specimens and quantum materials, exciting yet delicate samples that are currently difficult to investigate.