In large rotation analyses of beams and shells with displacement-based discretization, the iterative burden grows considerably with the membrane/flexural stiffness ratio. For linear elastic materials, it was shown that a stress-displacement iteration solves this issue and reduces the computational cost even of several times. Convergence problems occur also when large rotations are coupled to material nonlinearity, even if this case is not well addressed in the literature. The extension of the mixed iteration to large deformation problems with nonlinear constitutive laws is faced in this work, focusing on elastoplasticity. New iterative schemes are derived for both displacement-based finite elements (DFEs) and mixed finite elements (MFEs), each featuring further variables in the linearization along with displacements. For DFEs, it is shown that the most convenient approach is to impose the constitutive law for an independent integration point strain, with the strain-displacement compatibility solved together with the global equilibrium. For MFEs, as an extension of this strategy, the most performing scheme solves element compatibility and global equilibrium simultaneously, with the element state evaluated for independent strains work-conjugate of the stress DOFs. In both cases, global linear systems in displacements only are required and strain-driven material laws are easily considered. Numerous tests validate the proposed iterative schemes showing a significant reduction of the computational cost compared to standard approaches.