  We consider a resistive multi-fluid framework from the 3+1 space-time
foliation point-of-view, paying particular attention to issues relating to the
use of multi-parameter equations of state and the associated inversion from
evolved to primitive variables. We highlight relevant numerical issues that
arise for general systems with relative flows. As an application of the new
formulation, we consider a three-component system relevant for hot neutron
stars. In this case we let the baryons (neutrons and protons) move together,
but allow heat and electrons to exhibit relative flow. This reduces the problem
to three momentum equations; overall energy-momentum conservation, a
generalised Ohm's law and a heat equation. Our results provide a hierarchy of
increasingly complex models and prepare the ground for new state-of-the-art
simulations of relevant scenarios in relativistic astrophysics.
