  Vortex electron beams are freely propagating electron waves carrying
adjustable orbital angular momentum with respect to the propagation direction.
Such beams were experimentally realized just a few years ago and are now used
to probe various electromagnetic processes. So far, these experiments used the
single vortex electron beams, either propagating in external fields or
impacting a target. Here, we investigate the elastic scattering of two such
aligned vortex electron beams and demonstrate that this process allows one to
experimentally measure features which are impossible to detect in the usual
plane-wave scattering. The scattering amplitude of this process is well
approximated by two plane-wave scattering amplitudes with different momentum
transfers, which interfere and give direct experimental access to the Coulomb
phase. This phase (shift) affects the scattering of all charged particles and
has thus received significant theoretical attention but was never probed
experimentally. We show that a properly defined azimuthal asymmetry, which has
no counterpart in plane-wave scattering, allows one to directly measure the
Coulomb phase as function of the scattering angle.
