Summary: Let \(X\) be an \(n\)-dimensional, \(n\geq 3\), smooth projective variety defined over \(\mathbb{R}\) such that \(\text{Pic}(X)\cong\mathbb{Z}\) and \(H_1(X(\mathbb{C}),\mathbb{Z})= 0\), and \(d\geq 3\) an odd integer. Here we show how to construct a degree \(d\) finite morphism \(f: Y\to X\) defined over \(\mathbb{R}\) and such that \(Y\) has general type, \(\text{Pic}(Y)\cong\mathbb{Z}\), and no non-trivial automorphism.