Semi-supervised domain generalization (SSDG) in medical image segmentation offers a promising solution for generalizing to unseen domains during testing, addressing domain shift challenges and minimizing annotation costs. However, conventional SSDG methods assume labeled and unlabeled data are available for each source domain in the training set, a condition that is not always met in practice. The coexistence of limited annotation and domain shift in the training set is a prevalent issue. Thus, this paper explores a more practical and challenging scenario, cross-domain semi-supervised domain generalization (CD-SSDG), where domain shifts occur between labeled and unlabeled training data, in addition to shifts between training and testing sets. Existing SSDG methods exhibit sub-optimal performance under such domain shifts because of inaccurate pseudolabels. To address this issue, we propose a novel dual-supervised asymmetric co-training (DAC) framework tailored for CD-SSDG. Building upon the co-training paradigm with two sub-models offering cross pseudo supervision, our DAC framework integrates extra feature-level supervision and asymmetric auxiliary tasks for each sub-model. This feature-level supervision serves to address inaccurate pseudo supervision caused by domain shifts between labeled and unlabeled data, utilizing complementary supervision from the rich feature space. Additionally, two distinct auxiliary self-supervised tasks are integrated into each sub-model to enhance domain-invariant discriminative feature learning and prevent model collapse. Extensive experiments on real-world medical image segmentation datasets, i.e., Fundus, Polyp, and SCGM, demonstrate the robust generalizability of the proposed DAC framework.