  Many relativistic plasma environments in high-energy astrophysics, including
pulsar wind nebulae, hot accretion flows onto black holes, relativistic jets in
active galactic nuclei and gamma-ray bursts, and giant radio lobes, are
naturally turbulent. The plasma in these environments is often so hot that
synchrotron and inverse-Compton (IC) radiative cooling becomes important. In
this paper we investigate the general thermodynamic and radiative properties
(and hence the observational appearance) of an optically thin relativistically
hot plasma stirred by driven magnetohydrodynamic (MHD) turbulence and cooled by
radiation. We find that if the system reaches a statistical equilibrium where
turbulent heating is balanced by radiative cooling, the effective electron
temperature tends to attain a universal value $\theta = kT_e/m_e c^2 \sim
1/\sqrt{\tau_T}$, where $\tau_T=n_e\sigma_T L \ll 1$ is the system's Thomson
optical depth, essentially independent of the strength of turbulent driving or
magnetic field. This is because both MHD turbulent dissipation and synchrotron
cooling are proportional to the magnetic energy density. We also find that
synchrotron self-Compton (SSC) cooling and perhaps a few higher-order IC
components are automatically comparable to synchrotron in this regime. The
overall broadband radiation spectrum then consists of several distinct
components (synchrotron, SSC, etc.), well separated in photon energy (by a
factor $\sim \tau_T^{-1}$) and roughly equal in power. The number of IC peaks
is checked by Klein-Nishina effects and depends logarithmically on $\tau_T$ and
the magnetic field. We also examine the limitations due to synchrotron
self-absorption, explore applications to Crab PWN and blazar jets, and discuss
links to radiative magnetic reconnection.
