We present LMT/AzTEC 1.1mm observations of ∼100 luminous high-redshift dusty star-forming galaxy candidates from the ∼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 θFWHM ≈ 9.5 arcsec angular resolution, our observations reveal that at least 9 per cent of the targets break into multiple systems with SNR ≥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, IR 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 ultra-red sources at 4 < z < 6 of ∼3 × 10−7 Mpc−3 with a contribution to the obscured star-formation of ≳ 8 × 10−4 M⊙ yr−1 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 3mm 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.