Homeopathy is controversial and hotly debated. The conclusions of systematic reviews of randomised controlled trials of homeopathy vary from ‘comparable to conventional medicine’ to ‘no evidence of effects beyond placebo’. It is claimed that homeopathy conflicts with scientific laws and that homoeopaths reject the naturalistic outlook, but no evidence has been cited. We are homeopathic physicians and researchers who do not reject the scientific outlook; we believe that examination of the prior beliefs underlying this enduring stand-off can advance the debate. We show that interpretations of the same set of evidence—for homeopathy and for conventional medicine—can diverge. Prior disbelief in homeopathy is rooted in the perceived implausibility of any conceivable mechanism of action. Using the ‘crossword analogy’, we demonstrate that plausibility bias impedes assessment of the clinical evidence. Sweeping statements about the scientific impossibility of homeopathy are themselves unscientific: scientific statements must be precise and testable. There is growing evidence that homeopathic preparations can exert biological effects; due consideration of such research would reduce the influence of prior beliefs on the assessment of systematic review evidence.