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Given a ribbon graph $$\Gamma $$ with some extra structure, we define, using constructible sheaves, a dg category $$\mathrm {CPM}(\Gamma )$$ meant to model the Fukaya category of a Riemann surface in the cell of Teichmüller space described by $$\Gamma .$$ When $$\Gamma $$ is appropriately decorated and admits a combinatorial 'torus fibration with section,' we construct from $$\Gamma $$ a one-dimensional algebraic stack $$\widetilde{X}_\Gamma $$ with toric components. We prove that our model is equivalent to $$\mathcal {P}\mathrm {erf}(\widetilde{X}_\Gamma )$$ , the dg category of perfect complexes on $$\widetilde{X}_\Gamma $$ . [ABSTRACT FROM AUTHOR]