Smooth rational projective varieties with non-finitely generated discrete automorphism group and infinitely many real forms

We show, among other things, that for each integer $n \ge 3$, there is a smooth complex projective rational variety of dimension $n$, with discrete non-finitely generated automorphism group and with infinitely many mutually non-isomorphic real forms. Our result is inspired by the work of Lesieutre and the work of Dinh and Oguiso.
Title has been changed following referee. the final form to appear in Math. Ann