Lemma: Extension of the coefficient field

If $K\subseteq \mathbb{R}$ and $N,u$ have entries in $K$, ranks are unchanged on extending scalars to $\mathbb{R}$. A fit exists over $\mathbb{R}$ if and only if it exists over $K$.

Proof. Each minor vanishes over $K$ exactly when it vanishes over $\mathbb{R}$. Apply this to $N$ and $[N\mid u]$. Equality of their ranks is the consistency criterion over either field.