  We show that the ring of regular functions of every smooth affine log
Calabi-Yau surface with maximal boundary has a vector space basis parametrized
by its set of integer tropical points and a $\mathbb{C}$-algebra structure with
structure coefficients given by the geometric construction of Keel-Yu. To prove
this result, we first give a canonical compactification of the mirror family
associated with a pair $(Y,D)$ constructed by Gross-Hacking-Keel where $Y$ is a
smooth projective rational surface, $D$ is an anti-canonical cycle of rational
curves and $Y\setminus D$ is the minimal resolution of an affine surface with,
at worst, du Val singularities. Then, we compute periods for the compactified
family using techniques from work of Ruddat-Siebert and use this to give a
modular interpretation of the compactified mirror family.
