In computational pathology, deep learning (DL) models for tasks such as segmentation or tissue classification are known to suffer from domain shifts due to different staining techniques. Stain adaptation aims to reduce the generalization error between different stains by training a model on source stains that generalizes to target stains. Despite the abundance of target stain data, a key challenge is the lack of annotations. To address this, we propose a joint training between artificially labeled and unlabeled data including all available stained images called Unsupervised Latent Stain Adaptation (ULSA). Our method uses stain translation to enrich labeled source images with synthetic target images in order to increase the supervised signals. Moreover, we leverage unlabeled target stain images using stain-invariant feature consistency learning. With ULSA we present a semi-supervised strategy for efficient stain adaptation without access to annotated target stain data. Remarkably, ULSA is task agnostic in patch-level analysis for whole slide images (WSIs). Through extensive evaluation on external datasets, we demonstrate that ULSA achieves state-of-the-art (SOTA) performance in kidney tissue segmentation and breast cancer classification across a spectrum of staining variations. Our findings suggest that ULSA is an important framework for stain adaptation in computational pathology.