  In this paper we propose a way to construct an analytic space over a
non-archimedean field, starting with a real manifold with an affine structure
which has integral monodromy. Our construction is motivated by the junction of
Homological Mirror conjecture and geometric Strominger-Yau-Zaslow conjecture.
In particular, we glue from "flat pieces" an analytic K3 surface. As a
byproduct of our approach we obtain an action of an arithmetic subgroup of the
group $SO(1,18)$ by piecewise-linear transformations on the 2-dimensional
sphere $S^2$ equipped with naturally defined singular affine structure.
