We study the distribution and dynamics of the circum- and intergalactic medium using a dense galaxy survey covering the field around the Q0107 system, a unique z~1 projected quasar triplet. With full Ly$alpha$ coverage along all three lines-of-sight from z=0.18 to z=0.73, more than 1200 galaxy spectra, and two MUSE fields, we examine the structure of the gas around galaxies on 100-1000 kpc scales. We search for H I absorption systems occurring at the same redshift (within 500 $textrm{km}$ $textrm{s}^{-1}$) in multiple sightlines, finding with $>$ 99.9% significance that these systems are more frequent in the observed quasar spectra than in a randomly distributed population of absorbers. This is driven primarily by absorption with column densities N(H I) $> 10^{14}$ $textrm{cm}^{-2}$, whilst multi-sightline absorbers with lower column densities are consistent with a random distribution. Star-forming galaxies are more likely to be associated with multi-sightline absorption than quiescent galaxies. HST imaging provides inclinations and position angles for a subset of these galaxies. We observe a bimodality in the position angle of detected galaxy-absorber pairs, again driven mostly by high-column-density absorbers, with absorption preferentially along the major and minor axes of galaxies out to impact parameters of several hundred kpc. We find some evidence supporting a disk/outflow dichotomy, as H I absorbers near the projected major-axis of a galaxy show line-of-sight velocities that tend to align with the rotation of that galaxy, whilst minor-axis absorbers are twice as likely to exhibit O VI at the same redshift.