Lagrangian description of perfect fluids and modified gravity with an extra force

We revisit the issue of the correct Lagrangian description of a perfect fluid (${\mathcal{L}}_1=P$ versus ${\mathcal{L}}_2=-\rho$) in relation with modified gravity theories in which galactic luminous matter couples nonminimally to the Ricci scalar. These Lagrangians are only equivalent when the fluid couples minimally to gravity and not otherwise; in the presence of nonminimal coupling they give rise to two distinct theories with different predictions.