  In this article we provide a version of Chow's theorem for real analytic
Levi-flat hypersurfaces in the complex projective space $\mathbb{P}^{n}$, $n
\geq 2$. More specifically, we prove that a real analytic Levi-flat
hypersurface $M \subset \mathbb{P}^{n}$, with singular set of real dimension at
most $2n-4$ and whose Levi leaves are contained in algebraic hypersurfaces, is
tangent to the levels of a rational function in $\mathbb{P}^{n}$. As a
consequence, $M$ is a semialgebraic set. We also prove that a Levi foliation on
$\mathbb{P}^{n}$ - a singular real analytic foliation whose leaves are immersed
complex manifolds of codimension one - satisfying similar conditions - singular
set of real dimension at most $2n-4$ and all leaves algebraic - is defined by
the level sets of a rational function.
