Red, redder, reddest: SCUBA-2 imaging of colour-selected
  \textit{Herschel} sources
  High-redshift, luminous, dusty star forming galaxies (DSFGs) constrain the
extremity of galaxy formation theories. The most extreme are discovered through
follow-up on candidates in large area surveys. Here we present 850 $\mu$m
SCUBA-2 follow-up observations of 188 red DSFG candidates from the
\textit{Herschel} Multi-tiered Extragalactic Survey (HerMES) Large Mode Survey,
covering 274 deg$^2$. We detected 87 per cent with a signal-to-noise ratio $>$
3 at 850~$\mu$m. We introduce a new method for incorporating the confusion
noise in our spectral energy distribution fitting by sampling correlated flux
density fluctuations from a confusion limited map. The new 850~$\mu$m data
provide a better constraint on the photometric redshifts of the candidates,
with photometric redshift errors decreasing from $\sigma_z/(1+z)\approx0.21$ to
$0.15$. Comparison spectroscopic redshifts also found little bias ($\langle
(z-z_{\rm spec})/(1+z_{\rm spec})\rangle = 0.08 $). The mean photometric
redshift is found to be 3.6 with a dispersion of $0.4$ and we identify 21 DSFGs
with a high probability of lying at $z > 4$. After simulating our selection
effects we find number counts are consistent with phenomenological galaxy
evolution models. There is a statistically significant excess of WISE-1 and
SDSS sources near our red galaxies, giving a strong indication that lensing may
explain some of the apparently extreme objects. Nevertheless, our sample should
include examples of galaxies with the highest star formation rates in the
Universe ($\gg10^3$ M$_\odot$yr$^{-1}$).
