Singular integral-affine structures on S^2 arise naturally when studying K3 surfaces; algebraically, as the dual complex of the central fiber of a degeneration and symplectically, as the base of a Lagrangian fibration. The moduli space of such structures thus provides a bridge between moduli theory of algebraic and symplectic structures on K3 surfaces. In line with the Gross-Siebert program, this relationship evinces a program to extend the universal family of polarized K3 surfaces over a toroidal compactification.