Summary: Given a pairwise distance \(\boldsymbol{D}\) on the elements in a finite set \(\boldsymbol{X}\), the order distance \(\boldsymbol{\Delta}\boldsymbol{(D)}\) on \(\boldsymbol{X}\) is defined by first associating a total preorder \(\preceq_x\) on \(\boldsymbol{X}\) to each \(x \in \boldsymbol{X}\) based on \(\boldsymbol{D}\), and then quantifying the pairwise disagreement between these total preorders. The order distance can be useful in relational analyses because using \(\boldsymbol{\Delta}\boldsymbol{(D)}\) instead of \(D\) may make such analyses less sensitive to small variations in \(\boldsymbol{D}\). Relatively little is known about properties of \(\boldsymbol{\Delta}\boldsymbol{(D)}\) for general distances \(\boldsymbol{D}\). Indeed, nearly all previous work has focused on understanding the order distance of a treelike distance, that is, a distance that arises as the shortest path distances in a tree with non-negative edge weights and \(\boldsymbol{X}\) mapped into its vertex set. In this paper we study the order distance \(\boldsymbol{\Delta}\boldsymbol{(D)}\) for distances \(\boldsymbol{D}\) that can be decomposed into sums of simpler distances called split-distances. Such distances \(\boldsymbol{D}\) generalize treelike distances, and have applications in areas such as classification theory and phylogenetics.