  We propose an integrable bootstrap framework for the computation of
correlation functions for superstrings in $AdS_3\times S^3\times T^4$
backgrounds supported by an arbitrary mixture or Ramond-Ramond and
Neveu-Schwarz-Neveu-Schwarz fluxes. The framework extends the "hexagon
tessellation" approach which was originally proposed for $AdS_5\times S^5$ and
for the first time it demonstrates its applicability to other (less
supersymmetric) setups. We work out the hexagon form factor for two-particle
states, including its dressing factors which follow from those of the spectral
problem, and we show that it satisfies non-trivial consistency conditions. We
propose a bootstrap principle, slightly different from that of $AdS_5\times
S^5$, which allows to extend the form factor to arbitrarily many particles.
Finally, we compare its predictions with some correlation functions of
protected operators. Possible applications of this construction include the
study of wrapping corrections, of higher-point correlation functions, and of
non-planar corrections.
