  Observed scatter in the Lyman-alpha opacity of quasar sightlines at $z<6$ has
motivated measurements of the correlation between Ly$\alpha$ opacity and galaxy
density, as models that predict this scatter make strong and sometimes opposite
predictions for how they should be related. Our previous work associated two
highly opaque Ly$\alpha$ troughs at $z\sim5.7$ with a deficit of Lyman-$\alpha$
emitting galaxies (LAEs). In this work, we survey two of the most highly
transmissive lines of sight at this redshift, towards the $z=6.02$ quasar SDSS
J1306+0356 and the $z=6.17$ quasar PSO J359-06. We find that both fields are
underdense in LAEs within 10 $h^{-1}$ Mpc of the quasar sightline, somewhat
less extensive than underdensities associated with Ly$\alpha$ troughs. We
combine our observations with three additional fields from the literature, and
find that while fields with extreme opacities are generally underdense,
moderate opacities span a wider density range. The results at high opacities
are consistent with models that invoke UV background fluctuations and/or late
reionization to explain the observed scatter in IGM Ly$\alpha$ opacities. There
is tension at low opacities, however, as the models tend to associate lower IGM
Ly$\alpha$ opacities with higher densities. Although the number of fields
surveyed is still small, the low-opacity results may support a scenario in
which the ionizing background in low-density regions increases more rapidly
than some models suggest after becoming ionized. Elevated gas temperatures from
recent reionization may also be making these regions more transparent.
