  We present observations and analysis of the polarized radio emission from the
nearby radio galaxy Fornax A over 1.28--3.1 GHz, using data from the Australia
Telescope Compact Array (ATCA). In this, the first of two associated papers, we
use modern broadband polarimetric techniques to examine the nature and origin
of conspicuous low-polarization (low-$p$) patches in the lobes. We resolve the
low-$p$ patches, and find that their low fractional polarization is associated
with complicated frequency-dependent interference in the polarized signal
generated by Faraday effects along the line of sight. The low-$p$ patches are
spatially correlated with interfaces in the magnetic structure of the lobe,
across which the line-of-sight-projected magnetic field changes direction.
Spatial correlations with the sky-projected magnetic field orientation and
structure in total intensity are also identified and discussed. We argue that
the low-$p$ patches, along with associated reversals in the line-of-sight
magnetic field and other related phenomena, are best explained by the presence
of $\mathcal{O}(10^9)$ $M_\odot$ of magnetized thermal plasma in the lobes,
structured in shells or filaments, and likely advected from the ISM of NCG 1316
or its surrounding ICM. Our study underscores the power and utility of
spatially-resolved, broadband, full-polarization radio observations to reveal
new facets of flow behaviors and magneto-ionic structure in radio lobes and
their interplay with the surrounding environment.
