  We generalize the hard-thermal-loop effective action of the equilibrium
quark-gluon plasma to a non-equilibrium system which is space-time homogeneous
but for which the parton momentum distribution is anisotropic. We show that the
manifestly gauge-invariant Braaten-Pisarski form of the effective action can be
straightforwardly generalized and we verify that it then generates all n-point
functions following from collisionless gauge-covariant transport theory for a
homogeneous anisotropic plasma. On the other hand, the Taylor-Wong form of the
hard-thermal-loop effective action has a more complicated generalization to the
anisotropic case. Already in the simplest case of anisotropic distribution
functions, it involves an additional term that is gauge invariant by itself,
but nontrivial also in the static limit.
