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

Which choices of truth tables and consequence relations for two logics L₁ and L₂ ensure the satisfaction of the following split interpolation property: If two formulas φ and ψ share at least one propositional atom and φ classically entails ψ, then there is a formula χ that shares all its propositional atoms with both φ and ψ, such that φ entails χ in L₁ and χ entails ψ in L₂? 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.