  We propose an extension of the minimal-substitution prescription for coupling
the electromagnetic field to hadronic systems with internal structure. The
resulting rules of extended substitution necessarily distinguish between
couplings to scalar and Dirac particles. Moreover, they allow for the
incorporation of electromagnetic form factors for virtual photons in an
effective phenomenological framework. Applied to pions and nucleons, assumed to
be fully dressed to all orders, the resulting dressed currents are shown to be
locally gauge invariant. Moreover, half-on-shell expressions of (hadron
propagator)$\times$(electromagnetic current) needed in all descriptions of
physical processes will lose \textit{all} information about hadronic dressing
for real photons. The Ball-Chiu ansatz for the spin-1/2 current is seen to
suffer from an incomplete coupling procedure where some essential aspects of
the Dirac particle are effectively treated as those of a scalar particle.
Applied to real Compton scattering on pions and nucleons, we find that
\emph{all} dressing information cancels exactly when external hadrons are on
shell, leaving only gauge-invariant bare Born-type contributions with physical
masses. Hence, nontrivial descriptions necessarily require contact-type
two-photon processes obtained by hadrons looping around two photon insertion
points.
