  Previous analyses of various standard candles observed by the Gaia satellite
have reported statistically significant systematics in the parallaxes that have
improved from $\sim$250 $\mu$as in the first data release (DR1) to 50--80
$\mu$as in the second data release (DR2). Here we examine the parallaxes newly
reported in the Gaia early third data release (EDR3) using the same sample of
benchmark eclipsing binaries (EBs) we used to assess the DR1 and DR2
parallaxes. We find a mean offset of $-37\pm20$ $\mu$as (Gaia$-$EB), which
decreases to $-15\pm18$ $\mu$as after applying the corrections recommended by
the Gaia Mission team; global systematics in the Gaia parallaxes have clearly
improved and are no longer statistically significant for the EB sample, which
spans $5\lesssim{G}\lesssim12$ in brightness and 0.03--3 kpc in distance. We
also find that the RUWE goodness-of-fit statistic reported in Gaia EDR3 is
highly sensitive to unresolved companions (tertiaries in the case of our EB
sample) as well as to photocenter motion of the binaries themselves. RUWE is
nearly perfectly correlated ($r^2=0.82$) with photocenter motions down to
$\lesssim$0.1 mas, and surprisingly this correlation exists entirely within the
nominal "good" RUWE range of 1.0--1.4. This suggests that RUWE values even
slightly greater than 1.0 may signify unresolved binaries in Gaia, and that the
RUWE value can serve as a quantitative predictor of the photocenter motion.
