The non-reciprocity of a Sagnac interferometer provides ultra-high sensitivity for parameter estimation and offers a wide range of applications, especially for optical fiber sensing. In this work, we study a new type of non-reciprocity existing in an optical fiber Sagnac interferometer where the polarization dependent loss is taken into consideration. In particular, this non-reciprocity is irrelevant to the physical effects that have been considered in previous studies, which originates from the geometric phases induced by a continuous-weak-measurement. Thus, it has a unique phenomenon of sudden phase transition, which may open a new way for the future design of high precision optical fiber sensors.