The electrostatic complexation between DOTAP/DOPC unilamellar liposomes and an oppositely charged polyelectrolyte (NaPA) has been investigated in a wide range of the liposome surface charge density. We systematically characterized the reentrant condensation and the charge inversion of polyelectrolyte-decorated liposomes by means of dynamic light scattering and electrophoresis. We explored the stability of this model polyelectrolyte/colloid system at different values of the surface charge of the bare liposomes and by changing two independent control parameters of the suspensions: the polyelectrolyte/colloid charge ratio and the ionic strength of the aqueous suspending medium. The progressive addition of neutral DOPC lipid within the liposome membrane gave rise to an interesting phenomenon which has not been observed previously: the stability diagram of the suspensions showed a novel reentrance due to the crossing of the desorption threshold of the polyelectrolyte. Indeed, at fixed charge density of the bare DOTAP/DOPC liposomes and for a wide range of polyion concentrations, we showed that the simple electrolyte addition first (low salt regime) destabilizes the suspensions because of the enhanced screening of the residual repulsion between the complexes, and then (high salt regime) determines the onset of a new stable phase, originated by the absence of polyelectrolyte adsorption on the particle surfaces. We show that the observed phenomenology can be rationalized within the modified Velegol-Thwar model for heterogeneously charged particles and that the polyelectrolyte desorption fits well the predictions of the adsorption theory of Winkler and Cherstvy [1]. Our findings unambiguously support the picture of the reentrant condensation as driven by the correlated adsorption of the polyelectrolyte chains on the particle surface, providing interesting insights into possible mechanisms for tailoring complex colloids via salt-induced effects.