Finding suitably generic points on curves with an application to the construction of rigid real closed fields

Let $K$ be an algebraically closed field of characteristic 0 and transcendence degree at least 2. Let $C\subset K^2$ be an irreducible curve defined over $K$ but not defined over the algebraic closure of $\mathbb{Q}$. There is $(x,y)$ a $K$-point of $C$ such that $x$ and $y$ are algebraically independent. Moreover, if $C_0$ and $C_1$ are two such curves and there is a finite-to-finite algebraic correspondence between them defined over $K$, then there are corresponding $K$-points $(x_0,y_0)\in C_0$ and $(x_1,y_1)\in C_1$ such that $x_0$ and $y_0$ are algebraically independent and $x_1$ and $y_1$ are algebraically independent. We use the latter result to construct non-Archimedean real closed fields of transcendence degree $\kappa$ with no non-trivial automorphisms for all $2\le \kappa \le {\mathrm{\aleph }}_1$.