  We prove a Bonnet theorem for isometric immersions of submanifolds into the
products of an arbitrary number of simply connected real space forms. Then, we
prove the existence of associated families of minimal surfaces in such
products. Finally, in the case of $\mathbb{S}^2\times\mathbb{S}^2$, we give a
complex version of the main theorem in terms of the two canonical complex
structures of $\mathbb{S}^2\times\mathbb{S}^2$.
