  We modify the Einstein-Schrodinger theory to include a cosmological constant
$\Lambda_z$ which multiplies the symmetric metric, and we show how the theory
can be easily coupled to additional fields. The cosmological constant
$\Lambda_z$ is assumed to be nearly cancelled by Schrodinger's cosmological
constant $\Lambda_b$ which multiplies the nonsymmetric fundamental tensor, such
that the total $\Lambda=\Lambda_z+\Lambda_b$ matches measurement. The resulting
theory becomes exactly Einstein-Maxwell theory in the limit as
$|\Lambda_z|\to\infty$. For $|\Lambda_z|\sim 1/(Planck length)^2$ the field
equations match the ordinary Einstein and Maxwell equations except for extra
terms which are $<10^{-16}$ of the usual terms for worst-case field strengths
and rates-of-change accessible to measurement. Additional fields can be
included in the Lagrangian, and these fields may couple to the symmetric metric
and the electromagnetic vector potential, just as in Einstein-Maxwell theory.
The ordinary Lorentz force equation is obtained by taking the divergence of the
Einstein equations when sources are included. The Einstein-Infeld-Hoffmann
(EIH) equations of motion match the equations of motion for Einstein-Maxwell
theory to Newtonian/Coulombian order, which proves the existence of a Lorentz
force without requiring sources. This fixes a problem of the original
Einstein-Schrodinger theory, which failed to predict a Lorentz force. An exact
charged solution matches the Reissner-Nordstrom solution except for additional
terms which are $\sim 10^{-66}$ of the usual terms for worst-case radii
accessible to measurement. An exact electromagnetic plane-wave solution is
identical to its counterpart in Einstein-Maxwell theory.
