  We explicitly construct the fully back-reacted half-BPS solutions in Type IIB
supergravity which are dual to Wilson loops with 16 supersymmetries in
$\mathcal{N}=4$ 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 $AdS_2 \times S^2 \times S^4 \times \Sigma$, with isometry group
$SO(2,1)\times SO(3) \times SO(5)$ in terms of two locally harmonic functions
on a Riemann surface $\Sigma$ 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 $g \geq 0$ hyperelliptic surface $\Sigma$, all of
whose branch points lie on the real line. Each genus $g$ solution has only a
single asymptotic $AdS_5 \times S^5$ 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 $AdS_5 \times S^5$ with 3 free
parameters, while for genus $g \geq 1$, 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 $\Sigma$ 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.
