We solve a Bayesian inverse Reynolds-averaged Navier-Stokes (RANS) problem that assimilates mean flow data by jointly reconstructing the mean flow field and learning its unknown RANS parameters. We devise an algorithm that learns the most likely parameters of an algebraic effective viscosity model, and estimates their uncertainties, from mean flow data of a turbulent flow. We conduct a flow MRI experiment to obtain mean flow data of a confined turbulent jet in an idealized medical device known as the FDA (Food and Drug Administration) nozzle. The algorithm successfully reconstructs the mean flow field and learns the most likely turbulence model parameters without overfitting. The methodology accepts any turbulence model, be it algebraic (explicit) or multi-equation (implicit), as long as the model is differentiable, and naturally extends to unsteady turbulent flows.