An irreducible real analytic subvariety H of real dimension 2 n+ 1 in a complex manifold M is a Levi-flat subset if its regular part carries a complex foliation of dimension n. Locally, a germ of real analytic Levi-flat subset is contained in a germ of irreducible complex variety H ı of dimension n+ 1 , called intrinsic complexification, which can be globalized to a neighborhood of H in M provided H is a coherent analytic subvariety. In this case, a singular holomorphic foliation F of dimension n in M that is tangent to H is also tangent to H ı . In this paper, we prove integration results of local and global nature for the restriction to H ı of a singular holomorphic foliation F tangent to a real analytic Levi-flat subset H. From a local viewpoint, if n= 1 and H ı has an isolated singularity, then F|Hı has a meromorphic first integral. From a global perspective, when M= P N and H is coherent and of low codimension, H ı extends to an algebraic variety. In this case, F|Hı has a rational first integral provided infinitely many leaves of F in H are algebraic.