  We determine the fermionic collective modes of a quark-gluon plasma which is
anisotropic in momentum space. We calculate the fermion self-energy in both the
imaginary- and real-time formalisms and find that numerically and analytically
(for two special cases) there are no unstable fermionic modes. In addition we
demonstrate that in the hard-loop limit the Kubo-Martin-Schwinger condition,
which relates the off-diagonal components of the real-time fermion self-energy,
holds even for the anisotropic, and therefore non-equilibrium, quark-gluon
plasma considered here. The results obtained here set the stage for the
calculation of the non-equilibrium photon production rate from an anisotropic
quark-gluon plasma.
