Abelian tropical covers

Let 𝔄 be a finite abelian group. In this article, we classify harmonic 𝔄-covers of a tropical curve Γ (which allow dilation along edges and at vertices) in terms of the cohomology group of a suitably defined sheaf on Γ. We give a realizability criterion for harmonic 𝔄-covers by patching local monodromy data in an extended homology group on Γ. As an explicit example, we work out the case 𝔄=ℤ/pℤ and explain how realizability for such covers is related to the nowhere-zero flow problem from graph theory.