For the identification of a linear parameter varying (LPV) system steered by a scheduling variable evolving within a finite set, the local approach consists in separately estimating local linear time invariant (LTI) models corresponding to fixed values of the scheduling variable. It is shown in this paper that, without any global structural assumption of the considered LPV system, the local state-space LTI models do not contain the necessary information about the similarity transformations making them coherent. Nevertheless, it is possible to estimate these similarity transformations from input-output data under appropriate input excitation conditions. These estimations result in a common state basis of the transformed local LTI models, so that they form a coherent global LPV model, suitable for numerical simulations in the case of fast scheduling variable evolutions.