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This paper is the third installment in a series of papers devoted to thecomputation of enumerative invariants of abelian surfaces through the tropicalapproach. We develop a pearl diagram algorithm similar to the floor diagramalgorithm used in toric surfaces that concretely solves the tropical problem.  These diagrams can be used to prove specific cases of Oberdieck's multiplecover formula that reduce the computation of invariants for non-primitiveclasses to the primitive case, getting rid of all diagram considerations andproviding short explicit formulas. The latter can be used to prove thequasi-modularity of generating series of classical invariants, and thepolynomiality of coefficients of fixed codegree in the refined invariants.

Tropical curves in abelian surfaces III: pearl diagrams and multiple cover formulas