A low energy theory for nonlinear transport in finite-size single-wall carbon nanotubes, based on a microscopic model for the interacting p(z) electrons and successive bosonization, is presented. Due to the multiple degeneracy of the energy spectrum diagonal as well as off-diagonal (coherences) elements of the reduced density matrix contribute to the nonlinear transport. A four-electron periodicity with a characteristic ratio between adjacent peaks, as well as nonlinear transport features, in quantitative agreement with recent experiments, are predicted.