  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, and opens a new window to study particle acceleration. Both H.E.S.S. and
MAGIC described the observed $\gamma$-rays as arising from a single, external
shock. In this paper, 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, particularly the spectral shape and distinct light curve
peaks. Instead, we put forward a model involving multiple shocks that
reproduces the observed $\gamma$-ray spectrum and temporal evolution. The
simultaneous appearance of multiple distinct velocity components in the nova
optical spectrum over the first several days of the outburst supports the
presence of distinct shocks, which may arise either from the strong latitudinal
dependence of the density of the external circumbinary medium (e.g., in the
binary equatorial plane versus the poles) or due to internal collisions within
the white dwarf ejecta (as powers the $\gamma$-ray emission in classical
novae).
