Zero-shot image anomaly classification (AC) and segmentation (AS) are vital for industrial quality control, detecting defects without prior training data. Existing representation-based methods compare patch features with nearest neighbors in unlabeled test images but struggle with consistent anomalies -- similar defects recurring across multiple images -- resulting in poor AC/AS performance. We introduce Consistent-Anomaly Detection Graph (CoDeGraph), a novel algorithm that identifies and filters consistent anomalies from similarity computations. Our key insight is that normal patches in industrial images show stable, gradually increasing similarity to other test images, while consistent-anomaly patches exhibit abrupt similarity spikes after exhausting a limited set of similar matches, a phenomenon we term ``neighbor-burnout.'' CoDeGraph constructs an image-level graph, with images as nodes and edges connecting those with shared consistent-anomaly patterns, using community detection to filter these anomalies. We provide a theoretical foundation using Extreme Value Theory to explain the effectiveness of our approach. Experiments on MVTec AD with the ViT-L-14-336 backbone achieve 98.3% AUROC for AC and AS performance of 66.8% (+4.2%) F1 and 68.1% (+5.4%) AP over state-of-the-art zero-shot methods. Using the DINOv2 backbone further improves segmentation, yielding 69.1% (+6.5%) F1 and 71.9% (+9.2%) AP, demonstrating robustness across architectures.