Semi-supervised learning has a great potential in medical image segmentation tasks with a few labeled data, but most of them only consider single-modal data. The excellent characteristics of multi-modal data can improve the performance of semi-supervised segmentation for each image modality. However, a shortcoming for most existing multi-modal solutions is that as the corresponding processing models of the multi-modal data are highly coupled, multi-modal data are required not only in the training but also in the inference stages, which thus limits its usage in clinical practice. Consequently, we propose a semi-supervised contrastive mutual learning (Semi-CML) segmentation framework, where a novel area-similarity contrastive (ASC) loss leverages the cross-modal information and prediction consistency between different modalities to conduct contrastive mutual learning. Although Semi-CML can improve the segmentation performance of both modalities simultaneously, there is a performance gap between two modalities, i.e., there exists a modality whose segmentation performance is usually better than that of the other. Therefore, we further develop a soft pseudo-label re-learning (PReL) scheme to remedy this gap. We conducted experiments on two public multi-modal datasets. The results show that Semi-CML with PReL greatly outperforms the state-of-the-art semi-supervised segmentation methods and achieves a similar (and sometimes even better) performance as fully supervised segmentation methods with 100% labeled data, while reducing the cost of data annotation by 90%. We also conducted ablation studies to evaluate the effectiveness of the ASC loss and the PReL module.