Swift J1858.6-0814 is a transient neutron star X-ray binary discovered in 2018 October. Multiwavelength follow-up observations across the electromagnetic spectrum revealed many interesting properties, such as erratic flaring on minute time-scales and evidence for wind outflows at both X-ray and optical wavelengths, strong and variable local absorption, and an anomalously hard X-ray spectrum. Here, we report on a detailed radio observing campaign consisting of one observation at 5.5/9 GHz with the Australia Telescope Compact Array, and nine observations at 4.5/7.5 GHz with the Karl G. Jansky Very Large Array. A radio counterpart with a flat to inverted radio spectrum is detected in all observations, consistent with a compact jet being launched from the system. Swift J1858.6-0814 is highly variable at radio wavelengths in most observations, showing significant variability when imaged on 3-to-5-min time-scales and changing up to factors of 8 within 20 min. The periods of brightest radio emission are not associated with steep radio spectra, implying they do not originate from the launching of discrete ejecta. We find that the radio variability is similarly unlikely to have a geometric origin, be due to scintillation, or be causally related to the observed X-ray flaring. Instead, we find that it is consistent with being driven by variations in the accretion flow propagating down the compact jet. We compare the radio properties of SwiftJ1858.6-0814 with those of Eddington-limited X-ray binaries with similar X-ray and optical characteristics, but fail to find a match in radio variability, spectrum, and luminosity.