We present the results of a radio variability study of the gravitational lens PMN J1838-3427. Our motivation was to determine the Hubble constant by measuring the time delay between variations of the two quasar images. We monitored the system for 4 months (approximately 5 times longer than the expected delay) using the Australia Telescope Compact Array at 9 GHz. Although both images were variable on a timescale of a few days, no correlated intrinsic variability could be identified, and therefore no time delay could be measured. Notably, the fractional variation of the fainter image (8%) was greater than that of the brighter image (4%), whereas lensed images of a point source would have the same fractional variation. This effect can be explained, at least in part, as the refractive scintillation of both images due to the turbulent interstellar medium of the Galaxy.