  Self-consistent Hartree-Fock approximation combined with solutions of the
Bethe-Salpeter equation offers a powerful tool for studies of strong
correlation effects arising in condensed matter models, nuclear physics,
quantum field theories, and real materials. The standard finite-temperature
approach would be to first solve the problem in the Matsubara representation
and then apply numerical analytic continuation to the real-frequency axis to
link theoretical results with experimental probes, but this ill-conditioned
procedure often distorts important spectral features even for very accurate
imaginary-frequency data. We demonstrate that the ladder-type
finite-temperature Bethe-Salpeter equation in the Hartree-Fock basis for the
3-point vertex function and, ultimately, system's polarization can be
accurately solved directly on the real frequency axis using the diagrammatic
Monte Carlo technique and series resummation. We illustrate the method by
applying it to the homogeneous electron gas model and demonstrate how multiple
scattering events renormalize Landau damping.
