We present an algorithm for the computation of the topological type of a real compact Riemann surface associated to an algebraic curve, i.e., its genus and the properties of the set of fixed points of the anti-holomorphic involution, namely, the number of its connected components, and whether this set divides the surface into one or two connected components. This is achieved by transforming an arbitrary canonical homology basis to a homology basis where the A-cycles are invariant under the antiholomorphic involution.