  Protogalactic environments are typically identified using quasar absorption
lines, and these galactic building blocks can manifest as Damped Lyman-Alpha
Absorbers (DLAs) and Lyman Limit Systems (LLSs). We use radio observations of
Faraday effects to test whether DLAs and LLSs host a magnetised medium, by
combining DLA and LLS detections throughout the literature with 1.4 GHz
polarization data from the NRAO VLA Sky Survey (NVSS). We obtain a control, a
DLA, and a LLS sample consisting of 114, 19, and 27 lines-of-sight respectively
- all of which are polarized at $\ge8\sigma$ to ensure Rician bias is
negligible. Using a Bayesian framework, we are unable to detect either coherent
or random magnetic fields in DLAs: the regular coherent magnetic fields within
the DLAs must be $\le2.8$ $\mu$G, and the lack of depolarization is consistent
with the weakly magnetised gas in DLAs being non-turbulent and quiescent.
However, we find mild suggestive evidence that LLSs have coherent magnetic
fields: after controlling for the redshift-distribution of our data, we find a
71.5% probability that LLSs have a higher RM than a control sample. We also
find strong evidence that LLSs host random magnetic fields, with a 95.5%
probability that LLS lines-of-sight have lower polarized fractions than a
control sample. The regular coherent magnetic fields within the LLSs must be
$\le2.4$ $\mu$G, and the magnetised gas must be highly turbulent with a typical
scale on the order of $\approx5$-20 pc, which is similar to that of the Milky
Way. This is consistent with the standard dynamo pedagogy, whereby magnetic
fields in protogalaxies increase in coherence and strength as a function of
cosmic time. Our results are consistent with a hierarchical galaxy formation
scenario, with the DLAs, LLSs, and strong magnesium II (MgII) systems exploring
three different stages of magnetic field evolution in galaxies.
