Accreting neutron stars in low-mass X-ray binaries show outflows - and sometimes jets - in the general manner of accreting black holes. However, the quantitative link between the accretion flow (traced by X-rays) and outflows and/or jets (traced by radio emission) is much less well understood for neutron stars than for black holes, other than the general observation that neutron stars are fainter in the radio at a given X-ray luminosity. We use data from the deep MAVERIC radio continuum survey of Galactic globular clusters for a systematic radio and X-ray study of six luminous (L X > 1034 erg s-1) persistent neutron star X-ray binaries in our survey, as well as two other transient systems also captured by our data. We find that these neutron star X-ray binaries show an even larger range in radio luminosity than previously observed. In particular, in quiescence at L X ∼ 3 × 1034 erg s-1, the confirmed neutron star binary GRS 1747-312 in Terzan 6 sits near the upper envelope of the black hole radio/X-ray correlation, and the persistently accreting neutron star systems AC 211 (in M15) and X1850-087 (in NGC 6712) show unusual radio variability and luminous radio emission. We interpret AC 211 as an obscured "Z source"that is accreting at close to the Eddington limit, while the properties of X1850-087 are difficult to explain, and motivate future coordinated radio and X-ray observations. Overall, our results show that neutron stars do not follow a single relation between inflow and outflow, and confirm that their accretion dynamics are more complex than for black holes.