Homological mirror symmetry in the classical setting relates the bounded derived category of coherent sheaves on a Calabi-Yau manifold to the split-closure of the derived Fukaya category of the mirror Calabi-Yau.

In the paper 'Arithmetic mirror symmetry for the 2-torus', authors construct a $\mathbb{Z}$-linear equivalence between exact Fukaya category of a punctured torus and the category of perfect complexes of coherent sheaves on the central fiber of Tate curve.

Intuitively, this statement is only a half of the HMS conjecture since we also would like to have an equivalence between the 'Fukaya category of the central fiber of Tate curve' and the bounded derived category of coherent sheaves on the punctured elliptic curve.

The question is: is it possible to somehow define the notion of Fukaya category for the central fiber of the Tate curve (which is a curve in $\mathbb{P}^2(\mathbb{Z})$, not a smooth manifold)? If it is possible, can we construct an equivalence to the derived bounded category of coherent sheaves on the punctured elliptic curve?