Straightforward discretizations of chiral gauge theories lead to lattice actions S′ in which gauge invariance is broken. We describe a method to obtain an almost gauge invariant lattice action S from the action S′. The new action S is defined such that for all gauge fields on a gauge orbit it is the same (up to a global gauge transformation) as the action S′ evaluated for the gauge field fixed to a smooth gauge. We use the recently introduced Laplacian gauge to compute this gauge fixed gauge field without Gribov ambiguities. In a numerical simulation with S it is not necessary to fix the gauge, and hence gauge fixing terms and Fadeev-Popov ghosts are not required. A hybrid Monte Carlo algorithm for simulations with this new action is described.