  We show that the Nakai--Moishezon ampleness criterion holds for real line
bundles on complete schemes. As applications, we treat the relative
Nakai--Moishezon ampleness criterion for real line bundles and the
Nakai--Moishezon ampleness criterion for real line bundles on complete
algebraic spaces. The main ingredient of this paper is Birkar's
characterization of augmented base loci of real divisors on projective schemes.
