  We investigate type IIB strings on $\mathrm{AdS}_3\times \mathrm{S}^3\times
\mathrm{S}^3\times\mathrm{S}^1$ with mixed Ramond-Ramond (R-R) and
Neveu-Schwarz-Neveu-Schwarz (NS-NS) flux. By suitably gauge-fixing the closed
string Green-Schwarz (GS) action of this theory, we derive the off-shell
symmetry algebra and its representations. We use these to determine the
non-perturbative worldsheet S-matrix of fundamental excitations in the theory.
The analysis involves both massive and massless modes in complete generality.
The S-matrix we find involves a number of phase factors, which in turn satisfy
crossing equations that we also determine. We comment on the nature of the
heaviest modes of the theory, but leave their identification either as
composites or bound-states to a future investigation.
