ABSTRACT The extreme, time-variable Faraday rotation observed in the repeating fast radio burst (FRB) 121102 and its associated persistent synchrotron source demonstrates that some FRBs originate in dense, dynamic, and possibly relativistic magneto-ionic environments. Besides rotation of the linear polarization vector (Faraday rotation), such media can generally convert linear to circular polarization (Faraday conversion). We use non-detection of Faraday conversion, and the temporal variation in Faraday rotation and dispersion in bursts from FRB 121102 to constrain models where the progenitor inflates a relativistic nebula (persistent source) confined by a cold dense medium (e.g. supernova ejecta). We find that the persistent synchrotron source, if composed of an electron–proton plasma, must be an admixture of relativistic and non-relativistic (Lorentz factor γ < 5) electrons. Furthermore, we independently constrain the magnetic field in the cold confining medium, which provides the Faraday rotation, to be between 10 and 30 mG. This value is close to the equipartition magnetic field of the confined persistent source implying a self-consistent and overconstrained model that can explain the observations.