  This article studies the symplectic cohomology of affine algebraic surfaces
that admit a compactification by a normal crossings anticanonical divisor.
Using a toroidal structure near the compactification divisor, we describe the
complex computing symplectic cohomology, and compute enough differentials to
identify a basis for the degree-zero part of the symplectic cohomology. This
basis is indexed by integral points in a certain integral affine manifold,
providing a relationship to the theta functions of Gross--Hacking--Keel.
Included is a discussion of wrapped Floer cohomology of Lagrangian submanifolds
and a description of the product structure in a special case. We also show
that, after enhancing the coefficient ring, the degree--zero symplectic
cohomology defines a family degenerating to a singular surface obtained by
gluing together several affine planes.
