Summary: Let \({\mathcal C}\) be an affine or projective smooth real algebraic curve, that is, a smooth complex curve in \(\mathbb{A}^n_\mathbb{C}\) or \(\mathbb{P}^n_\mathbb{C}\) defined by real equations, having a non empty real part. Then every divisor \(E\) on \({\mathcal C}\), which is linearly equivalent to its conjugate \(E^c\), is also equivalent to a divisor supported on a set of real points of \({\mathcal C}\).