  We study the off-equilibrium dynamics of the Edwards-Anderson spin glass in
four dimensions under the influence of a non-hamiltonian perturbation. We find
that for small asymmetry the model behaves as the hamiltonian one, while for
large asymmetry the behaviour of the model can be well described by an
interrupted aging scenario. The autocorrelation function C(t_w+\tau,t_w) scales
as \tau/t_w^\beta, with \beta a function of the asymmetry. For very long
waiting times the previous regime crosses over to a time translational
invariant regime (TTI) with stretched exponential relaxation. The model does
not show signs of reaching a TTI regime for weak asymmetry, but in the aging
regime the exponent \beta is always different from one, showing a non trivial
aging scenario.
