  We consider a classical Two-Component Plasma analog of the Restricted
Primitive Model of electrolyte, where the hard-core interaction is replaced by
a soft differentiable potential. Within the Born-Green-Yvon hierarchy for the
equilibrium distribution functions, we shed light on an infection mechanism
where the charge correlations are polluted by the density correlations at the
critical point of the liquid-gas transition. This implies an algebraic decay of
critical charge correlations. Such breakdown of exponential clustering should
provide dielectric rather than conducting properties at the critical point,
leading to the violation of certain charge-charge sum rules. This is in
agreement with Monte Carlo simulations.
