The physical origin of hysteretic elasticity in consolidated granular media is still debated. We show that hysteresis in quasi-static experiments (slow loading/unloading cycles) and dynamic acoustoelastic testing (fast perturbation of the sample obtained by a propagating or standing wave) can be seen as a consequence of slow dynamics, which also induces elastic anisotropy due to nonlinearity. Both conditioning (i.e. the evolution of the elastic properties towards a non-equilibrium steady state when a load is applied) and relaxation (i.e. the slow recovery of the original elastic properties when the strain is reduced) are described as the consequence of a non-equilibrium strain generated in the material by the applied load. Experiments performed span over five orders of magnitude in strain (from 10-7 to 10-2) and we demonstrate the proposed model captures well the observed phenomenology in the full strain range considered.