Summary: We provide necessary and sufficient conditions for the uniqueness of minimisers of the Ginzburg-Landau functional for \(\mathbb{R}^n\)-valued maps under a suitable convexity assumption on the potential and for \(H^{1/2} \cap L^\infty\) boundary data that is non-negative in a fixed direction \(e\in \mathbb{S}^{n-1} \). Furthermore, we show that, when minimisers are not unique, the set of minimisers is generated from any of its elements using appropriate orthogonal transformations of \(\mathbb{R}^n\). We also prove corresponding results for harmonic maps with values into \(\mathbb{S}^{n-1}\).