In August of 2021, Fermi-LAT, H.E.S.S., and MAGIC detected GeV and TeV gamma-ray emission from an outburst of recurrent nova RS Ophiuchi. This detection represents the first very high energy gamma-rays observed from a nova, opening a new window onto particle acceleration and nova evolution. Both H.E.S.S. and MAGIC described the observed gamma-rays as arising from a single external shock. To better interpret this detection, we perform detailed, multi-zone modeling of RS Ophiuchi's 2021 outburst, including a self-consistent prescription for particle acceleration and magnetic field amplification. We demonstrate that, contrary to previous work, a single shock cannot simultaneously explain RS Ophiuchi's GeV and TeV emission. Instead, we put forward a model involving multiple shocks that reproduces the observed gamma-ray spectrum and temporal evolution. This result demonstrates the importance of modeling that properly accounts for the microphysics of particle acceleration when interpreting gamma-ray observations of novae and other astrophysical shocks.