  Einstein originally proposed a nonsymmetric tensor field, with its symmetric
part associated with the spacetime metric and its antisymmetric part associated
with the electromagnetic field, as an approach to a unified field theory. Here
we interpret it more modestly as an alternative to Einstein-Maxwell theory,
approximating the coupling between the electromagnetic field and spacetime
curvature in the macroscopic classical regime. Previously it was shown that the
Lorentz force can be derived from this theory, albeit with deviation on the
scale of a universal length constant $\ell$. Here we assume that $\ell$ is of
galactic scale and show that the modified coupling of the electromagnetic field
with charged particles allows a non-Maxwellian equilibrium of non-neutral
plasma. The resulting electromagnetic field is "dark" in the sense that its
modified Lorentz force on the plasma vanishes, yet through its modified
coupling to the gravitational field it engenders a nonvanishing, effective mass
density. We obtain a solution for which this mass density asymptotes
approximately to that of the pseudo-isothermal model of dark matter. The
resulting gravitational field produces radial acceleration, in the context of a
post-Minkowskian approximation, which is negligible at small radius but yields
a flat rotation curve at large radius. We further exhibit a family of such
solutions which, like the pseudo-isothermal model, has a free parameter to set
the mass scale (in this case related to the charge density) and a free
parameter to set the length scale (in this case an integer multiple of $\ell$).
Moreover, these solutions are members of a larger family with more general
angular and radial dependence. They thus show promise as approximations of
generalized pseudo-isothermal models, which in turn are known to fit a wide
range of mass density profiles for galaxies and clusters.
