  The evolution of an electrically conducting imcompressible fluid with
nonconstant density can be described by a set of equations combining the
continuity, momentum and Maxwell's equations; altogether known as the
inhomogeneous Navier--Stokes--Maxwell system.
  In this paper, we focus on the global well-posedness of these equations in
two dimensions. Specifically, we are able to prove the existence of global
energy solutions, provided that the initial velocity field belongs to the Besov
space $\dot{B}^{r}_{p,1}(\mathbb{R}^2)$, with $r=-1+\frac{2}{p}$, for some
$p\in (1,2)$, while the initial electromagnetic field enjoys some
$H^s(\mathbb{R}^2)$ Sobolev regularity, for some $s \geq 2-\frac{2}{p} \in
(0,1)$, and whenever the initial fluid density is bounded pointwise and close
to a nonnegative constant. Moreover, if it is assumed that $s>\frac{1}{2}$,
then the solution is shown to be unique in the class of all energy solutions.
  It is to be emphasized that the solutions constructed here are global and
uniformly bounded with respect to the speed of light $c\in (0,\infty)$. This
important fact allows us to derive the inhomogeneous MHD system as the speed of
light tends to infinity.
