We present LMT/AzTEC 1.1mm observations of $sim100$ luminous high-redshift dusty star-forming galaxy candidates from the $sim600,$sq.deg $Herschel$-ATLAS survey, selected on the basis of their SPIRE red far-infrared colours and with $S_{500murm m}=35-80$ mJy. With an effective $theta_{rm FWHM}approx9.5,$ arcsec angular resolution, our observations reveal that at least 9 per cent of the targets break into multiple systems with SNR $geq 4$ members. The fraction of multiple systems increases to $sim23,$ 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 $(z_{rm med}approx3.6)$, the redshift distribution of the latter is skewed towards higher redshifts. Of the AzTEC sources $sim85,$ per cent lie at $z_{rm phot}>3$ while $sim33,$ per cent are at $z_{rm phot}>4$. This corresponds to a lower limit on the space density of ultra-red sources at $4<z<6$ of $sim3times10^{-7}, textrm{Mpc}^{-3}$ with a contribution to the obscured star-formation of $gtrsim 8times10^{-4}, textrm{M}_odot textrm{yr}^{-1} textrm{Mpc}^{-3}$. 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 3mm LMT/RSR spectroscopic redshifts of six red-$Herschel$ galaxies at $z_{rm spec}=3.85-6.03$, two of them (at $z sim 4.7$) 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.
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