Terrestrial gamma ray flashes (TGFs) are a class of enigmatic electrical discharges in the Earth's atmosphere. In this study, we analyze an unprecedentedly large dataset comprised of 2188 TGFs whose signatures were simultaneously measured using space- and ground-based detectors over a five-year period. The Gamma-ray Burst Monitor (GBM) on board the Fermi spacecraft provided the energetic radiation measurements. Radio frequency (RF) measurements were obtained from the Global Lightning Dataset (GLD360). Here we show the existence of two categories of TGFs - those that were accompanied by quasi-simultaneous electromagnetic pulses (EMPs) detected by the GLD360 and those without such simultaneous EMPs. We examined, for the first time, the dependence of the TGF-associated EMP-peak-amplitude on the horizontal offset distance between the Fermi spacecraft and the TGF source. TGFs detected by the GBM with sources at farther horizontal distances are expected to be intrinsically brighter and were found to be associated with EMPs having larger median peak-amplitudes. This provides independent evidence that the EMPs and TGFs are produced by the same phenomenon, rather than the EMPs being from "regular" lightning in TGF-producing thunderstorms.