  Mach-Zehnder interferometry has been proposed as a probe for detecting the
statistics of anyonic quasiparticles in fractional quantum Hall (FQH) states.
Here we focus on interferometers made of multimode edge states with upstream
modes. We find that the interference visibility is suppressed due to
downstream-upstream mode entanglement; the latter serves as a "which path"
detector to the downstream interfering trajectories. Our analysis tackles a
concrete realization of filling factor 2/3, but its applicability goes beyond
that specific case, and encompasses the recent observation of ubiquitous
emergence of upstream neutral modes in FQH states. The latter, according to our
analysis, goes hand in hand with the failure to observe Mach-Zehnder anyonic
interference in fractional states. We point out how charge-neutral mode
disentanglement will resuscitate the interference signal.
