  We present a systematic shearing-box investigation of MRI-driven turbulence
in a weakly collisional plasma by including the effects of an anisotropic
pressure stress, i.e. anisotropic (Braginskii) viscosity. We constrain the
pressure anisotropy ($\Delta p$) to lie within the stability bounds that would
be otherwise imposed by kinetic microinstabilities. We explore a broad region
of parameter space by considering different Reynolds numbers and magnetic-field
configurations, including net vertical flux, net toroidal-vertical flux and
zero net flux. Remarkably, we find that the level of turbulence and
angular-momentum transport are not greatly affected by large anisotropic
viscosities: the Maxwell and Reynolds stresses do not differ much from the MHD
result. Angular-momentum transport in Braginskii MHD still depends strongly on
isotropic dissipation, e.g., the isotropic magnetic Prandtl number, even when
the anisotropic viscosity is orders of magnitude larger than the isotropic
diffusivities. Braginskii viscosity nevertheless changes the flow structure,
rearranging the turbulence to largely counter the parallel rate of strain from
the background shear. We also show that the volume-averaged pressure anisotropy
and anisotropic viscous transport decrease with increasing isotropic Reynolds
number (${\rm Re}$); e.g., in simulations with net vertical field, the ratio of
anisotropic to Maxwell stress ($\alpha_{\rm A} / \alpha_{\rm M}$) decreases
from $\sim 0.5$ to $\sim 0.1$ as we move from ${\rm Re} \sim 10^3$ to ${\rm Re}
\sim 10^4$. Anisotropic transport may thus become negligible at high ${\rm
Re}$. Anisotropic viscosity nevertheless becomes the dominant source of heating
at large ${\rm Re}$, accounting for $\gtrsim 50 \%$ of the plasma heating. We
conclude by briefly discussing the implications of our results for RIAFs onto
black holes.
