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The ideas of transversals are important to study algebraic structures which are useful for investigating semirings structure. In this paper, we introduce and explore split additively orthodox semirings which are special semirings with transversals. After obtaining some property theorems of split additively orthodox semirings, the structure theorem for them is obtained by using idempotent semirings, additively inverse semirings, and Munn semigroup. It not only extends and strengthens the corresponding results of Clifford semirings and split orthodox semigroups but also develops a new way to study semirings.