Experiments searching for Heavy Neutral Leptons (HNLs) typically interpret their results within simplified models consisting of a single HNL coupled to a single lepton flavor. However, any model which aims to describe neutrino oscillations necessarily features more than one HNL, coupled to several flavors. As we show in this work, the reinterpretation of the results of experimental searches in terms of realistic models is a non-trivial task. We perform a detailed reinterpretation of the latest ATLAS search for prompt HNLs in $W$ decays within a minimal low-scale seesaw with two HNLs. We show that the exclusion limits obtained using the detailed reinterpretation can differ by several orders of magnitude from the limits quoted for the simplified models. Hence naively comparing the mixing angles from a realistic model to the reported limits could lead to wrongly excluding entire regions of parameter space! To overcome this issue without requiring experiments to report constraints on all possible HNL models, we propose a simple framework that allows one to easily and accurately reinterpret exclusion limits within closely-related models. We outline a number of concrete steps that can be taken by experiments to implement this method with minimal effort, and we discuss its applicability to other models of feebly interacting particles.