  We present a theory of the transmission of incoherent guided matter-waves
through Sagnac interferometers. Interferometer configurations with only one
input and one output port have a property similar to the phase rigidity
observed in the transmission through Aharonov-Bohm interferometers in coherent
mesoscopic electronics. This property is connected to the existence of
counterpropagating paths of equal length and enables the operation of such
matter-wave interferometers with incoherent sources. High finesse
interferometers of this kind have a rotation sensitivity inversely proportional
to the square root of the finesse.
