In theories with chiral couplings, one of the important consistency requirements is that of the cancellation of a gauge anomaly. In particular, this is one of the conditions imposed on the hypercharges in the standard model. However, anomaly cancellation condition of the standard model looks unnatural from the perspective of a theory with extra dimensions. Indeed, if our world were embedded into an odd-dimensional space, then the full theory would be automatically anomaly-free. In this paper we discuss the physical consequences of anomaly noncancellation for effective 4-dimensional field theory. We demonstrate that in such a theory parallel electric and magnetic fields get modified. In particular, this happens for any particle possessing both electric charge and magnetic moment. This effect, if observed, can serve as a low energy signature of extra dimensions. On the other hand, if such an effect is absent or is very small, then from the point of view of any theory with extra dimensions it is just another fine-tuning and should acquire theoretical explanation. © 2005 The American Physical Society.