  Synchrotron emission from astrophysical nonthermal sources usually assumes
that the emitting particles are isotropic. By means of large-scale two- and
three-dimensional particle-in-cell simulations, we demonstrate that the
dissipation of magnetically-dominated ($\sigma_0\gg1$) turbulence in pair
plasmas leads to strongly anisotropic particle distributions. At Lorentz
factors $\sim \sigma_0 \gamma_{th0}$ (here, $\gamma_{th0}$ is the initial
Lorentz factor), the particle velocity is preferentially aligned with the local
magnetic field; instead, the highest energy particles are roughly isotropic.
This energy-dependent anisotropy leads to a synchrotron spectral flux $\nu
F_\nu\propto \nu^s$ that is much harder than for isotropic particles.
Remarkably, for $\sigma_0\gg1$ we find that the solid-angle-averaged spectral
slope in the slow cooling regime is $s\sim 0.5-0.7$ for a wide range of
turbulence fluctuations, $0.25\lesssim \delta B_{\rm rms0}^2/B_0^2\lesssim 4$,
despite significant variations in the power-law energy spectrum of nonthermal
particles. This is because weaker turbulence levels imprint a stronger degree
of anisotropy, thereby counteracting the effect of the steeper particle
spectrum. The synchrotron spectral slope may be even harder, $s\gtrsim 0.7$, if
the observer is in the plane perpendicular to the mean magnetic field. Our
results are independent of domain size and dimensionality. Our findings may
help explaining the origin of hard synchrotron spectra of astrophysical
nonthermal sources, most notably the radio spectrum of Pulsar Wind Nebulae.
