Consistency under paraphrase, the property that semantically equivalent prompts yield identical predictions, is increasingly used as a proxy for reliability when deploying medical vision-language models (VLMs). We show this proxy is fundamentally flawed: a model can achieve perfect consistency by relying on text patterns rather than the input image. We introduce a four-quadrant per-sample safety taxonomy that jointly evaluates consistency (stable predictions across paraphrased prompts) and image reliance (predictions that change when the image is removed). Samples are classified as Ideal (consistent and image-reliant), Fragile (inconsistent but image-reliant), Dangerous (consistent but not image-reliant), or Worst (inconsistent and not image-reliant). Evaluating five medical VLM configurations across two chest X-ray datasets (MIMIC-CXR, PadChest), we find that LoRA fine-tuning dramatically reduces flip rates but shifts a majority of samples into the Dangerous quadrant: LLaVA-Rad Base achieves a 1.5% flip rate on PadChest while 98.5% of its samples are Dangerous. Critically, Dangerous samples exhibit high accuracy (up to 99.6%) and low entropy, making them invisible to standard confidence-based screening. We observe a negative correlation between flip rate and Dangerous fraction (r = -0.89, n=10) and recommend that deployment evaluations always pair consistency checks with a text-only baseline: a single additional forward pass that exposes the false reliability trap.