  On a real regular elliptic surface without multiple fiber, the Betti number
$h_1$ and the Hodge number $h^{1,1}$ are related by $h_1\leq h^{1,1}$. We prove
that it's always possible to deform such algebraic surface to obtain
$h_1=h^{1,1}$. Furthermore, we can impose that each homology class can be
represented by a real algebraic curve. We use a real version of the modular
construction of elliptic surfaces.
