A real Jordan-curve symbol with a nonreal finite Toeplitz spectrum

The Laurent polynomial $b(z)=-64z^{-2}+8z^{-1}-128-8z-63z^2-16z^3-z^4$ is real on a star-shaped Jordan curve in $\mathbb C\setminus\{0\}$ enclosing zero, yet its $2\times2$ Toeplitz section has the nonreal eigenvalues $-128\pm8i$. The curve is $\{\rho(t)e^{it}\}$, where $\rho(t)\in(1/2,2)$ is the root of $r^2-1+\tfrac{r^3}{4}\cos t=0$, unique by strict monotonicity in $r$, so that $\rho$ is continuous; the polar form makes the curve injective. On this curve $a(z)=8z^{-1}+8z+z^2$ is real, hence so is $b=a-a^2$, whose extreme coefficients $b_{-2}=-64$ and $b_4=-1$ are nonzero. The mechanism is that polynomial composition preserves reality on the curve, whereas finite Toeplitz truncation does not commute with polynomial functional calculus. The argument concerns every finite section and makes no claim about the limiting spectrum.