I learned from this question and its answers that the imperfect marker -ba- comes from the same PIE root as fui and fio. What about the form fiebam (and other persons) then? Does it contain the same root twice (fi- and -ba-)?
If yes, how is this possible? It certainly does not look like reduplication. I imagine that the similarity of the two elements was forgotten and the imperfect form of fieri was introduced by analogy to the rest of Latin conjugation. Or perhaps it arose through periphrasis, with the same verb as the auxiliary and the main verb (cf. Italian ho avuto).
I find this a little confusing, so I would appreciate if someone could set me on the right track here.