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Robertson and Seymour proved two fundamental theorems about tangles ingraphs: the tree-of-tangles theorem, which says that every graph has atree-decomposition such that distinguishable tangles live in different nodes ofthe tree, and the tangle-tree duality theorem, which says that graphs without a$k$-tangle have a tree-decomposition that witnesses the non-existence of suchtangles, in that $k$-tangles would have to live in a node but no node is largeenough to accommodate one.  Erde combined these two fundamental theorems into one, by constructing asingle tree-decomposition such that every node either accommodates a single$k$-tangle or is too small to accommodate one. Such a tree-decomposition thusshows at a glance how many $k$-tangles a graph has and where they are.  The two fundamental theorems have since been extended to abstract separationsystems, which support tangles in more general discrete structures. In thispaper we extend Erde's unified theorem to such general systems.

Refining trees of tangles in abstract separation systems I: Inessential parts