Ibukiyama correspondences on automorphic forms on $\operatorname{Mp}_4(\mathbb{A}_\mathbb{Q})$ and $\operatorname{SO}_5(\mathbb{A}_\mathbb{Q})$ generating large discrete series representations at the real place

In our previous paper we gave proofs of Ibukiyama's correspondences on holomorphic Siegel modular forms of degree 2 of half-integral weight and integral weight. In this paper, we formulate and prove similar correspondences on automorphic forms on $\operatorname{Mp}_4(\mathbb{A}_\mathbb{Q})$ or $\operatorname{SO}_5(\mathbb{A}_\mathbb{Q})$ generating large discrete series representations at the real components. In addition, we show that the correspondences can be described in terms of local theta correspondences.