Type II Solutions on AdS$_3\times S^3\times S^3$ with Large Superconformal Symmetry

New local solutions in type II supergravity that are foliations of AdS$_3\times S^3\times S^3$ over an interval and preserve at least large $\mathcal{N}=(4,0)$ supersymmetry are found. Some cases have compact internal space, some not and one experiences an enhancement to $\mathcal{N}=(4,4)$. We present two new globally compact solutions with D brane and O plane sources explicitly, one in each of IIA and IIB. The former is part of an infinite family of solutions with D8/O8s back reacted on AdS$_3\times S^3\times S^3\times S^1$. In the latter the the algebra degenerates to small $\mathcal{N}=(4,0)$ and the internal geometry is bounded between D5s and O5s back reacted on AdS$_3\times S^3\times \mathbb{R}^4$.