Well-orders realized by CE equivalence relations

A structure $A$ is realized by an equivalence relation $E$ if there exists a structure $B$ such that $B/E$ is isomorphic to $A$. We will describe sets of ordinals that can be realized by one fixed computably enumerable equivalence relation, provided that this equivalence relation can realize an ordinal less than $\omega^\omega$.
(Joint work with N. A. Bazhenov.)