The class of real harmonizable fractional Lévy motions (RHFLMs) is introduced. It is shown that these share many properties with fractional Brownian motion. These fields are locally asymptotically self-similar with a constant index H, and have Hölderian paths. Moreover, the identification of H for the RHFLMs can be performed with the so-called generalized variation method. Besides fractional Brownian motion, this class contains non-Gaussian fields that are asymptotically self-similar at infinity with a real harmonizable fractional stable motion of index \tilde{H} as tangent field. This last property should be useful in modelling phenomena with multiscale behaviour.