  Recently, a finite-temperature real-time static potential has been introduced
via a Schr\"odinger-type equation satisfied by a certain heavy quarkonium
Green's function. Furthermore, it has been pointed out that it possesses an
imaginary part, which induces a finite width for the tip of the quarkonium peak
in the thermal dilepton production rate. The imaginary part originates from
Landau-damping of low-frequency gauge fields, which are essentially classical
due to their high occupation number. Here we show how the imaginary part can be
measured with classical lattice gauge theory simulations, accounting
non-perturbatively for the infrared sector of finite-temperature field theory.
We demonstrate that a non-vanishing imaginary part indeed exists
non-perturbatively; and that its value agrees semi-quantitatively with that
predicted by Hard Loop resummed perturbation theory.
