  We provide an off-shell formulation of four-dimensional higher spin gravity
based on a covariant Hamiltonian action on an open nine-dimensional Poisson
manifold whose boundary consists of the direct product of spacetime and a
noncommutative twistor space of S^2 x S^2 topology. The fundamental field is a
superconnection consisting of even and odd differential forms valued in the odd
and even sectors of a 3-graded associative algebra given by the direct product
of an eight-dimensional Frobenius algebra and a higher spin algebra extended by
inner Klein operators. The superconnection consists of two one-forms gauging
the one-sided actions of the higher spin algebra, two bi-fundamental real forms
given by the Weyl zero-form and a new dynamical two-form, an additional set of
forms providing a maximal duality extension, and, finally, the Lagrange
multipliers required for the covariant Hamiltonian action. In a particular
two-form background, the model yields Vasiliev's recently proposed extended
higher spin gravity equations, whose interaction terms are accounted for by de
Rham closed globally defined forms arising in the dynamical two-form.
