Multistage mass spectrometry (collecting multiple MS^3 spectra from each MS^2 spectrum) and accurate precursor masses (but inaccurate fragment masses) have been demonstrated to lead to significant gains in peptide identification via database search but have had a limited impact in de novo peptide sequencing. Our Multi-stage Spectral Networks package addresses both of these in a rigorous probabilistic framework for analyzing spectra of overlapping peptides, resulting in both accurate de novo peptide sequencing from multistage mass spectra (despite the inferior quality of MS^3 spectra) and improved interpretation of spectral networks. Additional details and the open-source package are available here.