Interstellar polycyclic aromatic hydrocarbons (PAHs) are strongly affected by the absorption of vacuum ultraviolet (VUV) photons in the interstellar medium (ISM), yet the branching ratio between ionization and fragmentation is poorly studied. This is crucial for the stability and charge state of PAHs in the ISM in different environments, affecting in turn the chemistry, the energy balance, and the contribution of PAHs to the extinction and emission curves. We studied the interaction of PAH cations with VUV photons in the 7-20 eV range from the synchrotron SOLEIL beamline, DESIRS. We recorded by action spectroscopy the relative intensities of photo-fragmentation and photo-ionization for a set of eight PAH cations ranging in size from 14 to 24 carbon atoms, with different structures. At photon energies below similar to 13.6 eV fragmentation dominates for the smaller species, while for larger species ionization is immediately competitive after the second ionization potential (IP). At higher photon energies all species behave similarly; the ionization yield gradually increases, leveling off between 0.8 and 0.9 at similar to 18 eV. Among isomers, PAH structure appears to mainly affect the fragmentation cross section but not the ionization cross section. We also measured the second IP for all species and the third IP for two of them; all are in good agreement with theoretical ones, confirming that PAH cations can be further ionized in the diffuse ISM. Determining actual PAH dication abundances in the ISM will require detailed modeling. Our measured photo-ionization yields for several PAH cations provide a necessary ingredient for such models.