Underwater environments impose severe challenges to visual-inertial odometry systems, as strong light attenuation, marine snow and turbidity, together with weakly exciting motions, degrade inertial observability and cause frequent tracking failures over long-term operation. While tightly coupled acoustic-visual-inertial fusion, typically implemented through an acoustic Doppler Velocity Log (DVL) integrated with visual-inertial measurements, can provide accurate state estimation, the associated graph-based optimization is often computationally prohibitive for real-time deployment on resource-constrained platforms. Here we present FAR-AVIO, a Schur-Complement based, tightly coupled acoustic-visual-inertial odometry framework tailored for underwater robots. FAR-AVIO embeds a Schur complement formulation into an Extended Kalman Filter(EKF), enabling joint pose-landmark optimization for accuracy while maintaining constant-time updates by efficiently marginalizing landmark states. On top of this backbone, we introduce Adaptive Weight Adjustment and Reliability Evaluation(AWARE), an online sensor health module that continuously assesses the reliability of visual, inertial and DVL measurements and adaptively regulates their sigma weights, and we develop an efficient online calibration scheme that jointly estimates DVL-IMU extrinsics, without dedicated calibration manoeuvres. Numerical simulations and real-world underwater experiments consistently show that FAR-AVIO outperforms state-of-the-art underwater SLAM baselines in both localization accuracy and computational efficiency, enabling robust operation on low-power embedded platforms. Our implementation has been released as open source software at https://far-vido.gitbook.io/far-vido-docs.