We develop algorithms which allow us to uniformize numerically any given real hyperelliptic plane M-curve. Starting from an equation for such a curve, we get (floating point approximations) for the generators of a discontinuous group of Möbius transformations uniformizing the given M-curve. Furthermore, we show how to compute an equation for a Riemann surface given by a such a group. For real hyperelliptic plane curves having maximal number of real components this construction gives a complete answer to the problem of numerical uniformization.