Medical referring image segmentation (MRIS) requires pixel-level masks aligned with textual descriptions of anatomical locations, making annotation costly in low-label regimes. Semi-supervised learning (SSL) can mitigate this burden by leveraging unlabeled data, but its success hinges on maintaining reliable image-text alignment under perturbations. Most existing SSL-based referred segmentation methods use either independent or simplistic multi-modal perturbations (e.g., left-right flips), without fully addressing cross-modal alignment under strong augmentation, while CutMix, highly effective in single-modal SSL, remains underexplored in multi-modal settings due to its tendency to disrupt image-text coherence. We propose Semi-MedRef, a teacher-student SSL framework designed to explicitly maintain consistency between medical images and positional language through three alignment-preserving components: T-PatchMix, a cross-modal CutMix-style augmentation that synchronizes patch mixing with referring expressions via position-constrained and probability-driven rules; PosAug, a position-aware text augmentation that masks or fuzzes anatomical phrases; and ITCL, a position-guided image-text contrastive learning module, which leverages positional pseudo-labels to construct soft anatomical positives and strengthen medically grounded cross-modal alignment. Experiments on QaTa-COV19 and MosMedData+ demonstrate that Semi-MedRef consistently outperforms both fully supervised and semi-supervised baselines across all label regimes.