  The worldsheet S matrix of strings on the $AdS_3\times S^3\times T^4$
background is almost entirely fixed by symmetries, up to five functions -- the
dressing factors. These must satisfy several consistency conditions, in
particular a set of crossing equations. We find that the existing proposal for
the dressing factors, while crossing invariant, violates some of these
consistency conditions. We put forward a new set of dressing factors and
discuss in detail their analytic properties in the string and mirror region, as
well as under bound-state fusion.
