  We establish a universality law for sequences of functions $\{w_n\}_{n \in
\mathbb{N}}$ satisfying a form of WKB approximation on compact intervals. This
includes eigenfunctions of generic Schr\"odinger operators, as well as Laguerre
and Chebyshev polynomials. Given two distinct points $x, y \in \mathbb{R}$, we
ask how often do $w_n(x)$ and $w_n(y)$ have the same sign. Asymptotically, one
would expect this to be true half the time, but this turns out to not always be
the case. Under certain natural assumptions, we prove that, for all $x \neq y$,
$$ \frac{1}{3} \leq \lim_{N \to \infty} \frac{1}{N} \# \left\{0 \leq n < N:
\mathrm{sgn}(w_n(x)) = \mathrm{sgn}(w_n(y)) \right\} \leq \frac{2}{3}, $$ and
that these bounds are optimal, and can be attained. Our methods extend to other
questions of similar flavor and we also discuss a number of open problems.
