Now for the 'almost all' bit. Some NeoDogs refuse to breed this way. They have become a separate people. Still NeoDogs, but refusing to play the past of the faithful companion to New Humans, insisting on breeding naturally, going their own way. This seventh non-breed are called - in accordance with the Tolkien theme - Wargs. Wargs are mutts - tending to the median, the wolfish unspecialized dog look. Wargs are fiercely independent, and make no compromises.
Here's a Terrier NeoDog for your enjoyment!