Split Interpolation: Refining Craig's Theorem via Three-Valued Logics

Which choices of truth tables and consequence relations for two logics $\mathsf{L}_1$ and $\mathsf{L}_2$ ensure the satisfaction of the following split interpolation property: If two formulas $\phi$ and $\psi$ share at least one propositional atom and $\phi$ classically entails $\psi$, then there is a formula $\chi$ that shares all its propositional atoms with both $\phi$ and $\psi$, such that $\phi$ entails $\chi$ in $\mathsf{L}_1$ and $\chi$ entails $\psi$ in $\mathsf{L}_2$? We identify the cases in which this property holds for any pair of propositional logics based on the same three-valued Boolean normal monotonic scheme for connectives and two monotonic consequence relations. Since the resulting logics are subclassical, every instance of this property constitutes a particular refinement of Craig's deductive interpolation theorem, as it entails the latter and further restricts the range of possible interpolants.