An intergalactic magnetic field (IGMF) stronger than 3× 10− 13 G would explain the lack of a bright, extended degree-scale, GeV-energy inverse Compton component in the gamma-ray spectra of TeV blazars. A robustly predicted consequence of the presence of such a field is the existence of degree-scale GeV-energy gamma-ray halos (gamma-ray bow ties) about TeV-bright active galactic nuclei, corresponding to more than half of all radio galaxies. However, the emitting regions of these halos are confined to and aligned with the direction of the relativistic jets associated with gamma-ray sources. Based on the orientation of radio jets, we align and stack corresponding degree-scale gamma-ray images of isolated Fanaroff–Riley class I and II objects and exclude the existence of these halos at overwhelming confidence, limiting the intergalactic field strength to< 10− 15 G for large-scale fields and progressively larger in the diffusive regime when the correlation length of the field becomes small in comparison to 1 Mpc. When combined with prior limits on the strength of the IGMF, this excludes a purely magnetic explanation for the absence of halos. Thus, it requires the existence of novel physical processes that preempt the creation of halos, eg, the presence of beam-plasma instabilities in the intergalactic medium or a drastic cutoff of the very high-energy spectrum of these sources.