IGR J17591−2342 is a new accreting millisecond X-ray pulsar (AMXP) that was recently discovered in outburst in 2018. Early observations revealed that the source's radio emission is brighter than that of any other known neutron star low-mass X-ray binary (NS-LMXB) at comparable X-ray luminosity, and assuming its likely ∼ > 6 kpc distance. It is comparably radio bright to black hole LMXBs at similar X-ray luminosities. In this work, we present the results of our extensive radio and X-ray monitoring campaign of the 2018 outburst of IGR J17591−2342. In total we collected 10 quasisimultaneous radio (VLA, ATCA) and X-ray (Swift-XRT) observations, which make IGR J17591−2342 one of the best-sampled NS-LMXBs. We use these to fit a power-law correlation index β = 0.37 +0.42 −0.40 between observed radio and X-ray luminosities (L R ∝ L X β ). However, our monitoring revealed a large scatter in IGR J17591−2342's radio luminosity (at a similar X-ray luminosity, L X ∼ 10 36 erg s −1 , and spectral state), with L R ∼ 4×10 29 erg s −1 during the first three reported observations, and up to a factor of 4 lower L R during later radio observations. Nonetheless, the average radio luminosity of IGR J17591−2342 is still one of the highest among NS-LMXBs, and we discuss possible reasons for the wide range of radio luminosities observed in such systems during outburst. We found no evidence for radio pulsations from IGR J17591−2342 in our Green Bank Telescope observations performed shortly after the source returned to quiescence. Nonetheless, we cannot rule out that IGR J17591−2342 becomes a radio millisecond pulsar during quiescence.