Quantum field theories with identical local dynamics can admit different choices of global structure, leading to different partition functions and spectra of extended operators. Recent work has determined the structure of such choices via geometric methods for various classes of non-Lagrangian theories obtained from stringy geometric engineering techniques. In this talk I will discuss a purely field theoretical counterpart of this analysis, showing that global structures can be captured from a careful analysis of the infrared Coulomb-like phases. Our results confirm and extend the many results obtained within geometric engineering about the global structures of Argyres-Douglas theories, 5d SCFTs and 6d SCFTs.