Indoor tracking and pose estimation, i.e., determining the position and orientation of a moving target, are increasingly important due to their numerous applications. While Inertial Navigation Systems (INS) provide high update rates, their positioning errors can accumulate rapidly over time. To mitigate this, it is common to integrate INS with complementary systems to correct drift and improve accuracy. This paper presents a novel approach that combines INS with an acoustic Riemannian-based localization system to enhance indoor positioning and orientation tracking. The proposed method employs both the Extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF) for fusing data from the two systems. The Riemannian-based localization system delivers high-accuracy estimates of the target's position and orientation, which are then used to correct the INS data. A new projection algorithm is introduced to map the EKF or UKF output onto the Riemannian manifold, further improving estimation accuracy. Our results show that the proposed methods significantly outperform benchmark algorithms in both position and orientation estimation. The effectiveness of the proposed methods was evaluated through extensive numerical simulations and testing using our in-house experimental setup. These evaluations confirm the superior performance of our approach in practical scenarios.