The scattering of plane sound waves incident in arbitrary directions on two elastic, fluid-filled, spherical shells in water is analyzed exactly. The shells can be close to each other, and thus can be strongly interacting. The analysis is performed in a broad frequency band that extends beyond the midfrequency (or ``coincidence'') enhancement region of the backscattered echoes. It is then possible to exactly account for the many, very complex features present at high frequencies within the scattering cross sections of multiply interacting shells. The incident and scattered wave fields are expanded in normal-mode series and the addition theorem for the spherical wave functions is used to determine the exact expression for the sound fields scattered by each shell in the presence---and repeatedly interacting with---the other, referred to as coordinate systems at the centers of either shell. Numerical evaluations involve the solution of a truncated, ill-conditioned, coupled, complex, transcendental, matrix system by means of the Gauss--Seidel iteration method. The large signature enhancement present for the end-fire configuration, above the coincidence region, is determined and explained as a novel elasto-structural acoustic effect caused by the focusing and reradiation of the a[inf 0] Lamb wave in the shells.