Summary: This paper is concerned with a phenomenological model of initially isotropic finite-strain multiplicative elasto-plasticity for polycrystals with grain boundary relaxation [the author, Contin. Mech. Thermodyn. 15, No. 2, 161--195 (2003; Zbl 1035.74015)]. We prove a local in time existence and uniqueness result for the corresponding initial-boundary value problem in the quasistatic rate-dependent case. Use is made of a generalized Korn first inequality [the author, Proc. R. Soc. Edinb., Sect. A, Math. 132, No. 1, 221--243 (2002; Zbl 1143.74311)] taking into account the incompatibility of the plastic deformation \(F_p\). This is a first result concerning classical solutions in geometrically exact nonlinear finite visco-plasticity for polycrystals. Global existence is not proved and cannot be expected due to the natural possibility of material degradation in time.