Think of the MT response as having two parts: an inductive part, which depends on how deep the fields diffuse and therefore changes with frequency/period, and a galvanic part, which depends only on the static charge distribution set up on the boundaries of near-surface conductivity anomalies.
The galvanic part arises purely from the requirement that current be conserved across an interface: wherever conductivity changes abruptly, charge accumulates on the boundary, and this charge generates its own static electric field. Because this charge configuration does not care about the period of the inducing field (it re-establishes itself essentially instantaneously each half-cycle), its effect on the electric field - and hence on apparent resistivity - is the same fraction at every frequency. On a log-log apparent resistivity plot this looks like a constant upward or downward shift of the whole curve, independent of period, which is exactly the "time-independent" separation the question describes.
Now, TE-mode apparent resistivity uses \(E\) measured along strike, while TM-mode apparent resistivity uses \(E\) measured across strike. A local 2-D or 3-D near-surface body distorts these two orthogonal electric-field components by different multiplicative factors (because the charge distribution on the body is not isotropic with respect to strike), so the static shifts applied to the TE and TM curves are different in size. This unequal, frequency-independent shift is what physically separates the two curves.
None of impedance phase, amplitude magnification or inducing frequency variation can explain a genuinely time-independent (frequency-independent) offset, since all three are properties of the inductive response, which by definition varies with period. Only galvanic distortion of the local electric field satisfies the "time-independent" condition stated in the question, confirming \(\boxed{\text{local distortion of electric field}}\) as correct, option (C).