Recent AI-generated image (AIGI) detectors perform well on natural-image benchmarks, but their behavior on text-rich forgeries, such as fabricated screenshots, documents, and news pages prevalent in misinformation, remains untested. We introduce TextFake, a 20,000-image benchmark for text-rich AIGI detection spanning 28 languages, 4 topic categories, and 2 scene modalities. Fake images are synthesized via a four-stage pipeline that annotates real images along three controlled dimensions and generates counterparts through distribution-aligned structured prompting, ruling out covariate shortcuts. Zero-shot evaluation of 14 specialized detectors and 3 frontier VLM APIs reveals a large systematic gap: no method exceeds 80% accuracy, with some dropping over 60% from natural-image benchmarks. Diagnostic evaluations identify three failure modes: the Text Density Curse, where dense glyphs overwhelm low-level detectors; Cloaking via Rendering Fidelity, where stronger text rendering suppresses enerative artifacts; and Threshold Collapse, where routine perturbations drive detectors toward chance-level performance.