We present Large Millimeter Telescope (LMT)/AzTEC 1.1 mm observations of ~100 luminous high-redshift dusty star-forming galaxy candidates from the $\sim 600\,$ sq.deg Herschel-ATLAS survey, selected on the basis of their SPIRE red far-infrared colours and with $S_{500\, \mu \rm m}=35-80$ mJy. With an effective $\theta _{\rm FWHM}\approx 9.5\,$arcsec angular resolution, our observations reveal that at least 9 per cent of the targets break into multiple systems with signal-to-noise ratio ≥4 members. The fraction of multiple systems increases to ~23 per cent (or more) if some non-detected targets are considered multiples, as suggested by the data. Combining the new AzTEC and deblended Herschel photometry, we derive photometric redshifts, infrared luminosities, and star formation rates. While the median redshifts of the multiple and single systems are similar (zmed ≍ 3.6), the redshift distribution of the latter is skewed towards higher redshifts. Of the AzTEC sources, ~85 per cent lie at zphot > 3 while ~33 per cent are at zphot > 4. This corresponds to a lower limit on the space density of ultrared sources at 4 < z < 6 of $\sim 3\times 10^{-7}\, \textrm {Mpc}^{-3}$ with a contribution to the obscured star formation of $\gtrsim 8\times 10^{-4}\, \textrm {M}_\odot \, \textrm {yr}^{-1} \, \textrm {Mpc}^{-3}$. Some of the multiple systems have members with photometric redshifts consistent among them suggesting possible physical associations. Given their angular separations, these systems are most likely galaxy over-densities and/or early-stage pre-coalescence mergers. Finally, we present 3 mm LMT/RSR spectroscopic redshifts of six red-Herschel galaxies at zspec = 3.85-6.03, two of them (at z ~ 4.7) representing new redshift confirmations. Here, we release the AzTEC and deblended Herschel photometry as well as catalogues of the most promising interacting systems and z > 4 galaxies.