Eukaryotes first: how could that be?

In the half century since the formulation of the prokaryote : eukaryote dichotomy, many authors have proposed that the former evolved from something resembling the latter, in defiance of common (and possibly common sense) views. In such 'eukaryotes first' (EF) scenarios, the last universal common ancestor is imagined to have possessed significantly many of the complex characteristics of contemporary eukaryotes, as relics of an earlier 'progenotic' period or RNA world. Bacteria and Archaea thus must have lost these complex features secondarily, through 'streamlining'. If the canonical three-domain tree in which Archaea and Eukarya are sisters is accepted, EF entails that Bacteria and Archaea are convergently prokaryotic. We ask what this means and how it might be tested.