Theories with Ehrenfeucht-Fraïssé equivalent non-isomorphic models

Our “long term and large scale” aim is to characterize the first order theories $T$ (at least the countable ones) such that for every ordinal $\alpha$ there are $\lambda$, $M_1$, $M_2$ such that $M_1$ and $M_2$ are non-isomorphic models of $T$ of cardinality $\lambda$ which are EF$^+_{\alpha,\lambda}$-equivalent. We expect that as in the main gap [11, XII], we get a strong dichotomy, i.e., on the non-structure side we have stronger, better examples, and on the structure side we have an analogue of [11, XIII]. We presently prove the consistency of the non-structure side for $T$ which is $\aleph_0$-independent (= not strongly dependent), even for PC$(T_1,T)$.