  We investigate n-gluon scattering amplitudes in the multi-Regge region of N=4
supersymmetric Yang-Mills theory at strong coupling. Through a careful analysis
of the thermodynamic bubble ansatz (TBA) for surfaces in AdS5 with n-g(lu)on
boundary conditions we demonstrate that the multi-Regge limit probes the large
volume regime of the TBA. In reaching the multi-Regge regime we encounter
wall-crossing in the TBA for all n>6. Our results imply that there exists an
auxiliary system of algebraic Bethe ansatz equations which encode valuable
information on the analytical structure of amplitudes at strong coupling.
