The predicted nature of the candidate redback pulsar 4FGL J2039.5−5617 was recently confirmed by the discovery of γ-ray millisecond pulsations (Clark et al. 2020, hereafter Paper I), which identify this γ-ray source as PSR J2039−5617. We observed 4FGL J2039.5−5617 with the Parkes radio telescope in 2016 and 2019. We detect radio pulsations at 1.4 GHz and 3.1 GHz, at the 2.6 ms period discovered in γ-rays, and also at 0.7 GHz in one 2015 archival observation. In all bands, the radio pulse profile is characterised by a single relatively broad peak which leads the main γ-ray peak. At 1.4 GHz we found clear evidence of eclipses of the radio signal for about half of the orbit, a characteristic phenomenon in redback systems, which we associate with the presence of intra-binary gas. From the dispersion measure DM= 24.57±0.03 pc cm −3 we derive a pulsar distance of 0.9±0.2 kpc or 1.7±0.7 kpc, depending on the assumed Galactic electron density model. Both the 1.4 GHz Parkes observations obtained in 2016 and 2019 and Swift monitoring campaign (2017)(2018) show no obvious long-term changes in the pulsar radio and X-ray emission. The modelling of the radio and γ-ray light curves leads to an independent determination of the orbital inclination, and to a determination of the pulsar mass, qualitatively consistent to the results in Paper I.