Versatile medical image segmentation (VMIS) targets the segmentation of multiple classes, while obtaining full annotations for all classes is often impractical due to the time and labor required. Leveraging partially labeled datasets (PLDs) presents a promising alternative; however, current VMIS approaches face significant class imbalance due to the unequal category distribution in PLDs. Existing methods attempt to address this by generating pseudo-full labels. Nevertheless, these typically require additional models and often result in potential performance degradation from label noise. In this work, we introduce a Task Consistency Training (TCT) framework to address class imbalance without requiring extra models. TCT includes a backbone network with a main segmentation head (MSH) for multi-channel predictions and multiple auxiliary task heads (ATHs) for task-specific predictions. By enforcing a consistency constraint between the MSH and ATH predictions, TCT effectively utilizes unlabeled anatomical structures. To avoid error propagation from low-consistency, potentially noisy data, we propose a filtering strategy to exclude such data. Additionally, we introduce a unified auxiliary uncertainty-weighted loss (UAUWL) to mitigate segmentation quality declines caused by the dominance of specific tasks. Extensive experiments on eight abdominal datasets from diverse clinical sites demonstrate our approach's effectiveness.