  We use Whitham deformations to give a complete account of spectral data of
real solutions of the sinh--Gordon equation of spectral genus 2. We
parameterise the closure of spectral data of constant mean curvature tori in
$\mathbb{R}^3$ by an isosceles right triangle and analyse its boundary. We
prove that the Wente family, which is described by spectral data with real
coefficients, is parameterised by the bisector of the right angle. Our methods
combine blowups of Whitham deformations and spectral data in an innovative way
that changes the underlying integrable system.
