Vision-language models (VLMs) often produce hallucinated outputs that are not supported by visual or textual evidence. While existing benchmarks have advanced hallucination evaluation, they primarily target object-level or task-specific errors and often assess different failure types on separate samples. This design limits direct comparison of hallucination patterns under shared evidence, and most prior evaluations remain English-centric. We introduce K-Prism, a Korean diagnostic benchmark for multi-faceted hallucination analysis in VLMs. K-Prism contains 1,002 multiple-choice questions across text and image tracks, organized into shared-evidence topic groups. Within each group, questions are designed to probe three diagnostic dimensions: factual, reasoning, and misleading. K-Prism uses a controlled dual-mode protocol comparing intact inputs with visually perturbed inputs to examine hallucination under reduced visual evidence. Experiments with closed- and open-source VLMs show dimension-specific degradation rather than a uniform accuracy drop. These findings demonstrate that hallucination in Korean VLMs cannot be captured by a single aggregate score and requires fine-grained diagnostic evaluation across evidence conditions and failure dimensions. Our dataset is publicly available at https://huggingface.co/spaces/KETI-NLP/K-Prism_Benchmark.