  We present the analysis of over 3,000 red-$Herschel$ sources
($S_{\mathrm{250\mu m}}<S_{\mathrm{350\mu m}}<S_{\mathrm{500\mu m} }$) using
public data from the ALMA archive and the $Herschel$-ATLAS survey. This
represents the largest sample of red-$Herschel$ sources with interferometric
follow-up observations to date. The high ALMA angular resolution and
sensitivity ($\theta_{\rm FWHM}\sim$1 arcsecond;
$\sigma_{1.3\mathrm{mm}}\sim0.17$ mJy beam$^{-1}$) allow us to classify the
sample into individual sources, multiple systems, and potential lenses and/or
close mergers. Interestingly, even at this high angular resolution, 73 per cent
of our detections are single systems, suggesting that most of these galaxies
are isolated and/or post-merger galaxies. For the remaining detections, 20 per
cent are classified as multiple systems, 5 per cent as lenses and/or mergers,
and 2 per cent as low-$z$ galaxies or Active Galactic Nuclei. Combining the
$Herschel$/SPIRE and ALMA photometry, these galaxies are found to be extreme
and massive systems with a median star formation rate of $\sim$ 1,500
$\mathrm{M_{\odot} yr^{-1}}$ and molecular gas mass of
$M_{\mathrm{gas}}\sim10^{11}$ $\mathrm{M_{\odot}}$. The median redshift of
individual sources is $z\approx2.8$, while the likely lensed systems are at
$z\approx3.3$, with redshift distributions extending to $z\sim6$. The inferred
depletion times are consistent with the starburst-type population with a mild
redshift evolution. Our results suggest a common star-formation mode for
extreme galaxies across cosmic time, likely triggered by close interactions or
disk-instabilities. Moreover, all galaxies with $S_{\mathrm{1.3mm}}\geq13$ mJy
are gravitationally amplified which, similar to the established $S_{500\mathrm{
\mu m}}>100$ mJy threshold, can be used as a simple criterion to identify
gravitationally lensed galaxies.
