  We present rest-frame optical spectra from the FMOS-COSMOS survey of twelve
$z \sim 1.6$ \textit{Herschel} starburst galaxies, with Star Formation Rate
(SFR) elevated by $\times$8, on average, above the star-forming Main Sequence
(MS). Comparing the H$\alpha$ to IR luminosity ratio and the Balmer Decrement
we find that the optically-thin regions of the sources contain on average only
$\sim 10$ percent of the total SFR whereas $\sim90$ percent comes from an
extremely obscured component which is revealed only by far-IR observations and
is optically-thick even in H$\alpha$. We measure the [NII]$_{6583}$/H$\alpha$
ratio, suggesting that the less obscured regions have a metal content similar
to that of the MS population at the same stellar masses and redshifts. However,
our objects appear to be metal-rich outliers from the metallicity-SFR
anticorrelation observed at fixed stellar mass for the MS population. The
[SII]$_{6732}$/[SII]$_{6717}$ ratio from the average spectrum indicates an
electron density $n_{\rm e} \sim 1,100\ \mathrm{cm}^{-3}$, larger than what
estimated for MS galaxies but only at the 1.5$\sigma$ level. Our results
provide supporting evidence that high-$z$ MS outliers are the analogous of
local ULIRGs, and are consistent with a major merger origin for the starburst
event.
