We analyze two-dimensional nonlinear sigma models at nonzero chemical potentials, which are governed by a complex action. In the spirit of contour deformations (thimbles), we extend the fields into the complex plane, which allows us to incorporate the chemical potentials mu as twisted boundary conditions. We write down the equations of motion and find exact BPS-like solutions in terms of pairs of (anti) holomorphic functions, in particular generalizations of unit charge and fractional instantons to generic mu. The decay of these solutions is controlled by the imaginary part of mu and a vanishing imaginary part causes jumps in the action. We analyze how the total charge is distributed into localized objects and to what extent these are characterized by topology.