Rotational inertial navigation systems (RINSs) are often employed in long-endurance navigation due to the high precision characteristics. Besides, redundant configuration is often adopted in practice to cope with potential faults. To effectively monitor the dual single-axis RINSs redundant system, this paper proposes a novel real-time fault monitoring algorithm that integrates adaptive filtering with parameter estimation. The adaptive fault monitoring filter based on state transformation error model with attitude observations increases the sensitivity of the filter to potential faults. Based on analysis of the filter estimation, a robust fault diagnosis criterion is proposed. Furthermore, the drift variation fitting mechanism is further proposed to address the challenge in monitoring faults of azimuth gyros. Experiments conclusively demonstrate the proficiency of the proposed method in achieving precise and swift fault monitoring.