  We use fully kinetic particle-in-cell simulations with unprecedentedly large
transverse box sizes to study particle acceleration in weakly-magnetized mildly
relativistic shocks traveling at a velocity $\approx 0.75c$ and a Mach number
of 15. We examine both subluminal (quasi-parallel) and superluminal
(quasi-perpendicular) magnetic field orientations. We find that quasi-parallel
shocks are mediated by a filamentary non-resonant (Bell) instability driven by
non-thermal ions, producing magnetic fluctuations on scales comparable to the
ion gyro-radius. In quasi-parallel shocks, both electrons and ions are
accelerated into non-thermal power-laws whose maximum energy grows linearly
with time. The upstream heating of electrons is small, and the two species
enter the shock front in rough thermal equilibrium. The shock's structure is
complex; the current of reflected non-thermal ions evacuates cavities in the
upstream which form filaments of amplified magnetic fields once advected
downstream. At late times, $10\%$ of the shock's energy goes into non-thermal
protons and $\gtrsim10\%$ into magnetic fields. We find that properly capturing
the magnetic turbulence driven by the non-thermal ions is important for
properly measuring the energy fraction of non-thermal electrons, $\epsilon_e$.
We find $\epsilon_e\sim 5\times10^{-4}$ for quasi-parallel shocks with
$v=0.75c$, slightly larger than what was measured in simulations of
non-relativistic shocks. In quasi-perpendicular shocks, no non-thermal
power-law develops in ions or electrons. The ion acceleration efficiency in
quasi-parallel shocks suggests that astrophysical objects that could host
mildly relativistic quasi-parallel shocks -- for example, the jets of active
galactic nuclei or microquasars -- may be important sources of cosmic rays and
their secondaries, such as gamma-rays and neutrinos.
