  Let $G/H$ be a semisimple symmetric space. Then the space $L^2(G/H)$ can be
decomposed into a finite sum of series representations induced from parabolic
subgroups of $G$. The most continuous part of the spectrum of $L^2(G/H)$ is the
part induced from the smallest possible parabolic subgroup. In this paper we
introduce Hardy spaces canonically related to this part of the spectrum for a
class of non-compactly causal symmetric spaces. The Hardy space is a
reproducing Hilbert space of holomorphic functions living on a tube type
bounded symmetric domain, containing $G/H$ as a boundary component. A boundary
value map is constructed and we show that it induces an $G$-isomorphism onto a
multiplicity free subspace of full spectrum in the most continuous part $L_{\rm
mc}^2(G/H)$ of $L^2(G/H)$. We also relate our Hardy space with the classical
Hardy space on the tube domain.
