  High-energy astrophysical systems frequently contain collisionless
relativistic plasmas that are heated by turbulent cascades and cooled by
emission of radiation. Understanding the nature of this radiative turbulence is
a frontier of extreme plasma astrophysics. In this paper, we use
particle-in-cell simulations to study the effects of external inverse Compton
radiation on turbulence driven in an optically thin, relativistic pair plasma.
We focus on the statistical steady state (where injected energy is balanced by
radiated energy) and perform a parameter scan spanning from low magnetization
to high magnetization ($0.04 \lesssim \sigma \lesssim 11$). We demonstrate that
the global particle energy distributions are quasi-thermal in all simulations,
with only a modest population of nonthermal energetic particles (extending the
tail by a factor of $\sim 2$). This indicates that nonthermal particle
acceleration (observed in similar non-radiative simulations) is quenched by
strong radiative cooling. The quasi-thermal energy distributions are well fit
by analytic models in which stochastic particle acceleration (due to, e.g.,
second-order Fermi mechanism or gyroresonant interactions) is balanced by the
radiation reaction force. Despite the efficient thermalization of the plasma,
nonthermal energetic particles do make a conspicuous appearance in the
anisotropy of the global momentum distribution as highly variable, intermittent
beams (for high magnetization cases). The beamed high-energy particles are
spatially coincident with intermittent current sheets, suggesting that
localized magnetic reconnection may be a mechanism for kinetic beaming. This
beaming phenomenon may explain rapid flares observed in various astrophysical
systems (such as blazar jets, the Crab nebula, and Sagittarius A*).
