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Random phase drift in single-mode optical fiber interferometers used with measurement systems, which is resulted from various types of environmental disturbances, should be eliminated in order to obtain high measurement precision. We propose an optical fiber interferometric measurement system which has the function of self-eliminating the random phase drift and is stable and robust enough for real-time precision measurement. By employing the characteristics of fiber Bragg gratings, the system interleaves two fiber Michelson interferometers together that share the common-interferometric-optical path. The signal of one of the interferometers is used to stabilize the system while the signal of the other interferometer is used for measurement. An electronic feedback loop for the stabilizing action is designed. The bandwidth of the feedback loop is 5 kHz, sufficiently wide to eliminate random phase drift resulted from various environmental disturbances. The system is endowed with high stability and therefore suitable for real-time precision measurement. By means of an active phase tracking technique to measure displacement, the linear regression coefficient of the displacement measurement results is 0.9998.