We explicitly construct the fully back-reacted half-BPS solutions in Type IIB supergravity which are dual to Wilson loops with 16 supersymmetries in N = 4 super Yang-Mills. In a first part, we use the methods of a companion paper to derive the exact general solution of the half-BPS equations on the space AdS₂ × S² × S⁴ × Σ, with isometry group SO(2, 1) × SO(3) × SO(5) in terms of two locally harmonic functions on a Riemann surface Σ with boundary. These solutions, generally, have varying dilaton and axion, and non-vanishing 3-form fluxes. In a second part, we impose regularity and topology conditions. These non-singular solutions may be parametrized by a genus g ≥ 0 hyperelliptic surface Σ, all of whose branch points lie on the real line. Each genus g solution has only a single asymptotic AdS₅ × S⁵ region, but exhibits g homology 3-spheres, and an extra g homology 5-spheres, carrying respectively RR 3-form and RR 5-form charges. For genus 0, we recover AdS₅ × S⁵ with 3 free parameters, while for genus g ≥ 1, the solution has 2g+5 free parameters. The genus 1 case is studied in detail. Numerical analysis is used to show that the solutions are regular throughout the g = 1 parameter space. Collapse of a branch cut on Σ subtending either a homology 3-sphere or a homology 5-sphere is non-singular and yields the genus g−1 solution. This behavior is precisely expected of a proper dual to a Wilson loop in gauge theory.