  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 $S_{500\mu\rm m}=35-80$ 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
$\sim3\times10^{-7}\, \textrm{Mpc}^{-3}$ with a contribution to the obscured
star-formation of $\gtrsim 8\times10^{-4}\, \textrm{M}_\odot \textrm{yr}^{-1}
\textrm{Mpc}^{-3}$. 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.
