  Phase sensitivity and thermal dephasing in coherent electron transport in
quasi one-dimensional (1D) waveguide rings of an asymmetric four-terminal
geometry are studied by magnetotransport measurements. We demonstrate the
electrostatic control of the phase in Aharonov-Bohm (AB) resistance
oscillations and investigate the impact of the measurement circuitry on
decoherence. Phase rigidity is broken due to the ring geometry: Orthogonal
waveguide cross-junctions and 1D leads minimize reflections and resonances
between leads allowing for a continuous electron transmission phase shift. The
measurement circuitry influences dephasing: Thermal averaging dominates in the
non-local measurement configuration while additional influence of potential
fluctuations becomes relevant in the local configuration.
