Let (X, μ) be a space with a finite measure μ, let A and B be w*-closed subalgebras of L^∞(μ), and let C and D be closed subspaces of L^p(μ) (1 <p < ∞) that are modules over A and B, respectively. Under certain additional assumptions, the couple (C ⋂ D,C ⋂ D ⋂ L^∞(μ)) is K-closed in (L^p(μ), L^∞(μ)). This statement covers, in particular, two cases analyzed previously: that of Hardy spaces on the two-dimensional torus and that of the coinvariant subspaces of the shift operator on the circle. Furthermore, many situations when A and B are w*-Dirichlet algebras also fit in this pattern.