Nakai--Moishezon ampleness criterion for real line bundles

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.