Terrestrial gamma ray flashes (TGFs) are beams of high‐energy photons associated to lightning. These photons are the bremsstrahlung of energetic electrons whose origin is currently explained by two mechanisms: energizing electrons in weak, but large‐scale thundercloud fields or the acceleration of low‐energy electrons in strong, but localized fields of lightning leaders. Contemporary measurements by the Atmosphere‐Space Interactions Monitor suggest that the production of TGFs is related to the leader step and associated streamer coronae when upward moving intracloud lightning illuminates. Based on these observations, we apply a particle‐in‐cell Monte Carlo code tracing electrons in the superposed electric field of two encountering streamer coronae and modeling the subsequent photon emission. We also perform a parameter study by solving the deterministic equations of motion for one electron. We find that this mechanism can explain the occurrence of TGFs with photons energies of several MeV lasting for tens to hundreds of μs, in agreement with observations. Plain Language Summary For more than two decades, it has been known that thunderstorms emit high‐energy X‐rays and γ rays, the so‐called terrestrial gamma ray flashes (TGFs) lasting for tens to hundreds of μs, which are the bremsstrahlung (“braking radiation”) of energetic electrons and are the most energetic natural phenomena on Earth. Within the last years, two theories have been crystallized out to explain the origin of energetic electrons: the acceleration and multiplication of energetic electrons as remnants of cosmic rays in the large‐scale electric fields of thunderclouds or the acceleration of thermal electrons in high electric fields in the vicinity of the tips of lightning leaders. Contemporary measurements of the Atmosphere‐Space Interactions Monitor (ASIM) show that TGFs are produced at the onset of the main optical lightning pulse, indicating that the electron acceleration is related to the upward pointing lighting leader tip. We have performed computational simulations of the electron acceleration in the superposed electric field of two encountering streamer coronae, a compilation of small plasma channels with high‐field tips, arising in the proximity of the lightning leader tip and the upper charge layer. We find that this scenario can explain the occurrence of TGFs with energies and durations compatible with previous and contemporary measurements. Key Points Relativistic electrons are produced during the breakdown of ICs during a current surge when two streamer coronae approach each other The acceleration of electrons between two streamer coronae leads to TGFs lasting for tens to hundreds of μs with photon energies of O(10 MeV) The maximum photon energy in TGFs is determined by the electric field of the upper cloud charge layer