  We study the moving phase of two-dimensional (2D) incompressible polar active
fluids in the presence of both quenched and annealed disorder. We show that
long-range polar order persists even in this defect-ridden two-dimensional
system. We obtain the large-distance, long-time scaling laws of the velocity
fluctuations using three distinct dynamic renormalization group schemes. These
are an uncontrolled one-loop calculation in exactly two dimensions, and two
$d=(d_c-\epsilon)$-expansions to $O(\epsilon)$, obtained by two different
analytic continuations of our 2D model to higher spatial dimensions: a ``hard"
continuation which has $d_c={7\over 3}$, and a ``soft" continuation with
$d_c={5\over 2}$. Surprisingly, the quenched and annealed parts of the velocity
correlation function have the same anisotropy exponent and the relaxational and
propagating parts of the dispersion relation have the same dynamic exponent in
the nonlinear theory even though they are distinct in the linearized theory.
This is due to anomalous hydrodynamics. Furthermore, all three renormalization
schemes yield very similar values for the universal exponents, and, therefore,
we expect the numerical values we predict for them to be highly accurate.
