Full-waveform inversion (FWI) based on the minimization of data residuals may not enhance our understanding of the subsurface and can at times lead to misleading subsurface models. Additionally, unconstrained multiparameter FWI may also lead to models that do not represent realistic lithology for independently derived parameters. We have developed a method for elastic FWI that explicitly imposes petrophysical restrictions to guide models toward realistic and feasible lithology, that is, to subsurface models consistent with the seismic data and with the underlying petrophysics. We exploit petrophysical information, such as that provided by well logs, to constrain the inversion and to avoid implausible models. We achieve this goal by confining the inverted models to a feasible region defined by a probability density function instead of imposing lithologic facies as a function of position. Inside this feasible petrophysical regime, the inverted models do not need to obey a specific trend, that is, we do not link the parameters with explicit and potentially inaccurate petrophysical relations. Instead, we define a petrophysical basin of attraction that confines models to a feasible region validated by regional lithologic and petrophysical information. We find through elastic models that incorporating probabilistic petrophysical constraints into the inversion objective function leads to models that are superior to models obtained either without constraints or with approximate analytic constraints. In addition, we discover that these constraints can help in mitigating common issues affecting elastic FWI, such as the artifacts produced by interparameter crosstalk and limited acquisition coverage.