  In this paper we consider the collective modes of a momentum-space
anisotropic quark-gluon plasma taking into account the effect of collisions
between the plasma constituents. Our analysis is carried out using a
collisional kernel of Bhatnagar-Gross-Krook form and extends prior analyses in
the literature by considering all possible angles of propagation of the gluonic
modes relative to the momentum-anisotropy axis. We extract both the stable and
unstable modes as a function of the collision rate and confirm prior findings
that gluonic unstable modes can be eliminated from the spectrum if the
collision rate is sufficiently large. In addition, we discuss the conditions
necessary for the existence of unstable modes and present evidence that
unstable mode growth rates are maximal for modes with momentum along the
anisotropy direction. Finally, we demonstrate that when there is a finite
collisional rate, gluonic unstable modes are absent from the spectrum at both
small and large momentum anisotropy. These results pave the way for
understanding the impact of collisions on a variety of non-equilibrium
quark-gluon plasma observables.
