The interpretation of LHC data, and the assessment of possible hints of new physics, requires the precise knowledge of the proton subnuclear structure in terms of parton distribution functions (PDFs). In this talk, I present a systematic methodology designed to determine whether and how global PDF fits might inadvertently ‘fit away’ signs of new physics in the high-energy tails of the distributions.I showcase a scenario, in which the PDFs may completely absorb such signs of new physics, thus biasing theoretical predictions and their interpretation. I discuss strategies to single out the effects in this scenario and disentangle the inconsistencies that stem from them. Finally, I will present a new tool that is able to perform simultaneous fits of PDFs and SMEFT Wilson coefficients that might help disentangling the effects highlighted in this talk.