  Real-time classical SU($N$) gauge theories at non-zero temperature contain
linear divergences. We introduce counterterms for these divergences in the
equations of motion in the continuum and on the lattice. These counterterms can
be given in terms of auxiliary fields that satisfy local equations of motion.
We present a lattice model with 6+1D auxiliary fields that for IR-sensitive
quantities yields cut-off independent results to leading order in the coupling.
Also an approximation with 5+1D auxiliary fields is discussed.
