Iterative-based methods have become mainstream in stereo matching due to their high performance. However, these methods heavily rely on labeled data and face challenges with unlabeled real-world data. To this end, we propose a consistency-aware self-training framework for iterative-based stereo matching for the first time, leveraging real-world unlabeled data in a teacher-student manner. We first observe that regions with larger errors tend to exhibit more pronounced oscillation characteristics during model prediction.Based on this, we introduce a novel consistency-aware soft filtering module to evaluate the reliability of teacher-predicted pseudo-labels, which consists of a multi-resolution prediction consistency filter and an iterative prediction consistency filter to assess the prediction fluctuations of multiple resolutions and iterative optimization respectively. Further, we introduce a consistency-aware soft-weighted loss to adjust the weight of pseudo-labels accordingly, relieving the error accumulation and performance degradation problem due to incorrect pseudo-labels. Extensive experiments demonstrate that our method can improve the performance of various iterative-based stereo matching approaches in various scenarios. In particular, our method can achieve further enhancements over the current SOTA methods on several benchmark datasets.