  We discuss the resummation of the strong coupling asymptotic expansion of the
dressing phase of the ${\rm AdS}_5\times {\rm S}^5$ superstring. The dressing
phase proposed by Beisert, Eden and Staudacher can be recovered from a modified
Borel-Ecalle resummation of this asymptotic expansion only by completing it
with new, non-perturbative and exponentially suppressed terms that can be
organized into different sectors labelled by an instanton-like number. We
compute the contribution to the dressing phase coming from the sum over all the
instanton sectors and show that it satisfies the homogeneous crossing symmetry
equation. We comment on the semiclassical origin of the non-perturbative terms
from the world-sheet theory point of view even though their precise explanation
remains still quite mysterious.
