Health-related misinformation claims often falsely cite a credible biomedical publication as evidence, which superficially appears to support the false claim. The publication does not really support the claim, but a reader could believe it thanks to the use of logical fallacies. Here, we aim to detect and to highlight such fallacies, which requires carefully assessing the exact content of the misrepresented publications. To achieve this, we introduce MissciPlus, an extension of the fallacy detection dataset Missci. MissciPlus builds on Missci by grounding the applied fallacies in real-world passages from misrepresented studies. This creates a realistic test-bed for detecting and verbalizing these fallacies under real-world input conditions, and enables novel passage-retrieval tasks. MissciPlus is the first logical fallacy dataset which pairs the real-world misrepresented evidence with incorrect claims, identical to the input to evidence-based fact-checking models. With MissciPlus, we i) benchmark retrieval models in identifying passages that support claims only when fallacies are applied, ii) evaluate how well LLMs articulate fallacious reasoning from misrepresented scientific passages, and iii) assess the effectiveness of fact-checking models in refuting claims that misrepresent biomedical research. Our findings show that current fact-checking models struggle to use relevant passages from misrepresented publications to refute misinformation. Moreover, these passages can mislead LLMs into accepting false claims as true.