Let $(G,G^\prime)$ be the reductive dual pair $(Sp(n,\Bbb R), O(k))$ or $(U(p,q),U(k))$, and let $K$ be a maximal compact subgroup of the noncompact group $G$. Then for the representations $\pi$ of $\widetilde{G}$ which occur in the Howe duality correspondence for $(G, G^\prime)$, we construct explicit intertwining maps between mixed models of $\pi$ and spaces of holomorphic sections of vector bundles over the hermitian symmetric space $G/K$, where $G/K$ is embedded in its holomorphic tangent space as a type III Siegel domain. This result provides a link between the original construction of these representations using tube domain and type II Siegel domain realizations of $G/K$ and more recent constructions using the bounded domain realization of $G/K$.