Non-perturbative BRST invariance and what it might be good for

We construct a local, gauge-fixed, lattice Yang-Mills theory with an exact BRST invariance, and with the same perturbative expansion as the standard Yang-Mills theory. The ghost sector, and some of its BRST transformation rules, are modified to get around Neuberger's theorem. A special term is introduced in the action to regularize the Gribov horizons, and the limit where the regulator is removed is discussed. We conclude with a few comments on what might be the physical significance of this theory. We speculate that there may exist new strong-interaction phases apart from the anticipated confinement phase.