  Quantum field theories with identical local dynamics can admit different
choices of global structure, leading to different partition functions and
spectra of extended operators. Such choices can be reformulated in terms of a
topological field theory in one dimension higher, the symmetry TFT. In this
paper we show that this TFT can be reconstructed from a careful analysis of the
infrared Coulomb-like phases. In particular, the TFT matches between the UV and
the IR. This provides a purely field theoretical counterpart of several recent
results obtained via geometric engineering in various string/M/F theory setups
for theories in four and five dimensions that we confirm and extend.
