We have obtained follow-up observations of the quasar pair LBQS 0107-025A,B and new observations of the nearby quasar LBQS 0107-0232 with the Hubble Space Telescope Faint Object Spectrograph. Extended wavelength coverage of LBQS 0107-025A and B using the G270H grating was also obtained. This triple system is unique in providing sensitivity to coherent Ly-alpha absorption on transverse scales of approximately 1 Mpc at z < 1. Monte Carlo simulations were used to establish the confidence level for matches between absorption features in different lines of sight as a function of velocity separation. Pairwise, there are 8, 9, and 12 lines that match between spectra. Three instances of matches between all three lines of sight were found with velocity separations of less than 550 kms^-1. Two of the pairings have coincident lines within |Delta v| < 200 kms^-1 that would occur with less than 10% probability by chance. Taking into account the equivalent widths of the lines, one of these triple coincidences is significant at the 99.99% confidence level, based on Monte Carlo simulations with random line placements. Matches with strong lines preferentially have small velocity separations. These same simulations are used to demonstrate that the distribution of matches for a population of absorbers randomly distributed in velocity space is peaked toward |Delta v| = 0, which has implications for the statistical significance of matches. One of the triple coincidences appears to be a strong absorber with a sheet-like, but inhomogeneous, geometry and a coherence length approaching or exceeding 1 Mpc.