We report a gravitational B F-type action principle propagating two (complex) degrees of freedom that, besides the gauge connection and the B field, only employs an additional Lagrange multiplier. The action depends on two parameters and remarkably is polynomial in the B field. For a particular choice of the involved parameters the action provides an alternative description of (complex) general relativity with a nonvanishing cosmological constant, whereas another choice corresponds to anti-self-dual gravity. Generic values of the parameters produce “close neighbors” of general relativity, although there is a peculiar choice of the parameters that leads to a Hamiltonian theory with two scalar constraints. Given the nontrivial form of the resulting scalar constraint for these models, we consider a more general setting where the scalar constraint is replaced with an arbitrary analytic function of some fundamental …