  We report new radio observations of a sample of thirty-six neutron star (NS)
X-ray binaries, more than doubling the sample in the literature observed at
current-day sensitivities. These sources include thirteen weakly-magnetised
($B<10^{10}$ G) and twenty-three strongly-magnetised ($B\geq10^{10}$ G) NSs.
Sixteen of the latter category reside in high-mass X-ray binaries, of which
only two systems were radio-detected previously. We detect four weakly and nine
strongly-magnetised NSs; the latter are systematically radio fainter than the
former and do not exceed $L_R \approx 3\times10^{28}$ erg/s. In turn, we
confirm the earlier finding that the weakly-magnetized NSs are typically radio
fainter than accreting stellar-mass black holes. While an unambiguous
identification of the origin of radio emission in high-mass X-ray binaries is
challenging, we find that in all but two detected sources (Vela X-1 and 4U
1700-37) the radio emission appears more likely attributable to a jet than the
donor star wind. The strongly-magnetised NS sample does not reveal a global
correlation between X-ray and radio luminosity, which may be a result of
sensitivity limits. Furthermore, we discuss the effect of NS spin and magnetic
field on radio luminosity and jet power in our sample. No current model can
account for all observed properties, necessitating the development and
refinement of NS jet models to include magnetic field strengths up to $10^{13}$
G. Finally, we discuss jet quenching in soft states of NS low-mass X-ray
binaries, the radio non-detections of all observed very-faint X-ray binaries in
our sample, and future radio campaigns of accreting NSs.
