  We describe a class of unified theories of electromagnetism and gravity. The
Lagrangian is of the BF type, with a potential for the B-field, the gauge group
is U(2) (complexified). Given a choice of the potential function the theory is
a deformation of (complex) general relativity and electromagnetism, and
describes just two propagating polarisations of the graviton and two of the
photon. When gravity is switched off the theory becomes the usual non-linear
electrodynamics with a general structure function. The Einstein-Maxwell theory
can be recovered by sending some of the parameters of the defining potential to
zero, but for any generic choice of the potential the theory is
indistinguishable from Einstein-Maxwell at low energies. A real theory is
obtained by imposing suitable reality conditions. We also study the
spherically-symmetric solution and show how the usual Reissner-Nordstrom
solution is recovered.
