  We show homological mirror symmetry results relating coherent analytic
sheaves on some complex elliptic surfaces and objects of certain Fukaya
categories. We first define the notion of a non-algebraic Landau-Ginzburg model
on $\mathbb{R} \times S^1$ and its associated Fukaya category, and show that
non-K\"{a}hler sur-faces obtained by performing two logarithmic transformations
to the product of the projective plane and an elliptic curve have non-algebraic
Landau-Ginzburg models as their mirror spaces; this class of surface includes
the classical Hopf surface $S^1\times S^3$ and other elliptic primary and
secondary Hopf surfaces. We also define localization maps from the Fukaya
categories associated to the Landau-Ginzburg models to partially wrapped and
fully wrapped categories. We show mirror symmetry results that relate the
partially wrapped and fully wrapped categories to spaces of coherent analytic
sheaves on open submanifolds of the compact complex surfaces in question, and
we use these results to sketch a proof of a full HMS result.
