Multidetection (MD) systems are characterized by multiple observation modes (OMs), and hence, simultaneously produce multiple measurements for each target. The key challenge in exploiting MD systems for multitarget tracking (MTT), compared to single-detection (SD) systems, is the significant amount of extra computational burden involved in order to solve the resulting multidimensional assignment problem among measurements, targets, and OMs. This article presents a novel computationally efficient MTT framework for MD systems, wherein the multitarget state is modeled as a random finite set (RFS), and a bank of OM-dependent MTT RFS filters with SD model are employed to recursively provide OM-dependent posteriors. The latter, which contain both real and false targets, are then suitably fused so as to enhance consensus on the true targets while weakening trust on the existence of the false ones. In this way, the computational complexity is significantly reduced compared to existing MTT algorithms with the MD model. Two representative RFS filters, i.e., unlabeled probability hypothesis density (PHD) and labeled multi-Bernoulli (LMB), are considered in the proposed framework and the computational complexity of the resulting MD MTT algorithms is analyzed. Performance of the proposed approach is assessed by simulation experiments in both over-the-horizon-radar (OTHR) and single-frequency-network passive radar (SFN-PR) MTT applications.