  In this paper we study flat deformations of real subschemes of
$\mathbb{P}^n$, hyperbolic with respect to a fixed linear subspace, i.e.
admitting a finite surjective and real fibered linear projection. We show that
the subset of the corresponding Hilbert scheme consisting of such subschemes is
closed and connected in the classical topology. Every smooth variety in this
set lies in the interior of this set. Furthermore, we provide sufficient
conditions for a hyperbolic subscheme to admit a flat deformation to a smooth
hyperbolic subscheme. This leads to new examples of smooth hyperbolic
varieties.
