  Using hard-loop (HL) effective theory for an anisotropic non-Abelian plasma,
which even in the static limit involves nonvanishing HL vertices, we calculate
the imaginary part of the static next-to-leading-order gluon self energy in the
limit of a small anisotropy and with external momentum parallel to the
anisotropy direction. At leading order, the static propagator has space-like
poles corresponding to plasma instabilities. On the basis of a calculation
using bare vertices, it has been conjectured that, at next-to-leading order,
the static gluon self energy acquires an imaginary part which regulates these
space-like poles. We find that the one-loop resummed expression taken over
naively from the imaginary-time formalism does yield a nonvanishing imaginary
part even after including all HL vertices. However, this result is not correct.
Starting from the real-time formalism, which is required in a non-equilibrium
situation, we construct a resummed retarded HL propagator with correct
causality properties and show that the static limit of the retarded
one-loop-resummed gluon self-energy is real. This result is also required for
the time-ordered propagator to exist at next-to-leading order.
