This paper proposes an Adaptive Tandem Kalman Filter (ATKF) algorithmic method for accurate, robust, and magnetometer-free heading estimation through the competitive fusion of inertial and positioning data. It aims to mitigate the angular heading drift, typically present in MEMS IMU (Micro-Electromechanical Systems Inertial Measurement Unit) sensors without the use of environmentally sensitive magnetometers. A series of simulated comparison tests with one of the state-of-the-art algorithms have demonstrated the high stability and robustness of the proposed algorithm. It has shown a consistent 40% to 90% improvement in the estimated heading accuracy and precision, depending on the maneuvering intensity and the data quality. Simulation results were experimentally validated during the full-scale test campaign, conducted in the industrial environment using a highly maneuverable forklift. Real-time forklift heading was tracked by the proposed ATKF algorithm with 1 degree overall median error and 2.3 degrees median error during the active movement periods. The proposed method has respectively shown 95% and 93% improvement in initial IMU heading accuracy and precision. Experimental evaluation of the magnetometer performance has practically confirmed its unreliability and inconsistency for industrial applications. The proposed ATKF heading estimation algorithm may find a variety of possible applications in the field of robotics and intelligent vehicles. It is expected to be especially useful in magnetometer-denied environments.