In this study we deal with the nonlinear Riemann--Hilbert problem (in short (RHP)) for generalized analytic functions in multiply connected domains. Using a similarity principle for multiply connected domains (presented here for the first time), we reduce the nonlinear RHP for generalized analytic functions to a corresponding nonlinear RHP for holomorphic functions with Holder continuous boundary data. Then the Newton--Kantorovic method combined with a continuation procedure as well as a new existence theorem for holomorphic solutions, which is based on topological degree arguments, leads to existence of at least two topologically different generalized analytic functions solving the nonlinear RHP.