We address the recently observed unexpected behavior of Aharonov-Bohm oscillations in the electronic Mach-Zehnder interferometer that was realized experimentally in a quantum Hall system [I. Neder et al., Phys. Rev. Lett. 96, 016804 (2006)]. We argue that the measured lobe structure in the visibility of oscillations and the phase rigidity result from a strong long-range interaction between two adjacent counterpropagating edge states, which leads to a resonant scattering of plasmons. The visibility and phase shift, which we express in terms of the transmission coefficient for plasmons, can be used for the tomography of edge states.