Three kinds of cells exist with increasingly complex membrane-protein targeting: Unibacteria (Archaebacteria, Posibacteria) with one cytoplasmic membrane (CM); Negibacteria with a two-membrane envelope (inner CM; outer membrane [OM]); eukaryotes with a plasma membrane and topologically distinct endomembranes and peroxisomes. I combine evidence from multigene trees, palaeontology, and cell biology to show that eukaryotes and archaebacteria are sisters, forming the clade neomura that evolved similar to 1.2 Gy ago from a posibacterium, whose DNA segregation and cell division were destabilized by murein wall loss and rescued by the evolving novel neomuran endoskeleton, histones, cytokinesis, and glycoproteins. Phagotrophy then induced coevolving serial major changes making eukaryote cells, culminating in two dissimilar cilia via a novel gliding-fishing-swimming scenario. I transfer Chloroflexi to Posibacteria, root the universal tree between them and Heliobacteria, and argue that Negibacteria are a clade whose OM, evolving in a green posibacterium, was never lost.