  Strong singly-ionized magnesium (MgII) absorption lines in quasar spectra
typically serve as a proxy for intervening galaxies along the line of sight.
Previous studies have found a correlation between the number of these MgII
absorbers and the Faraday rotation measure (RM) at $\approx5$ GHz. We
cross-match a sample of 35,752 optically-identified non-intrinsic MgII
absorption systems with 25,649 polarized background radio sources for which we
have measurements of both the spectral index and RM at 1.4 GHz. We use the
spectral index to split the resulting sample of 599 sources into flat-spectrum
and steep-spectrum subsamples. We find that our flat-spectrum sample shows
significant ($\sim3.5\sigma$) evidence for a correlation between MgII
absorption and RM at 1.4 GHz, while our steep-spectrum sample shows no such
correlation. We argue that such an effect cannot be explained by either
luminosity or other observational effects, by evolution in another confounding
variable, by wavelength-dependent polarization structure in an active galactic
nucleus, by the Galactic foreground, by cosmological expansion, or by partial
coverage models. We conclude that our data are most consistent with intervenors
directly contributing to the Faraday rotation along the line of sight, and that
the intervening systems must therefore have coherent magnetic fields of
substantial strength ($\bar{B}=1.8\pm0.4$ $\mu$G). Nevertheless, the weak
nature of the correlation will require future high-resolution and broadband
radio observations in order to place it on a much firmer statistical footing.
