by Dave Koski with vZome
Whereas I've made much about the Mite, as dissected by Fuller into A & B modules, as the most primitive space-filling tetrahedron without overt handedness, we do have another contender, just by those simple criteria (tetrahedron, no handedness).
Because of its all-isosceles triangular facets, the Rite's component quarter tetrahedra (apex at the center of gravity) may all be rotated into one another, and so are minus overt chirality.
Assembled from two Mites, the Rite is itself a space-filler, as is the Bite (the other of two tetrahedral Sytes). These quarter Rites would have a volume of 1/16, same as a half-Mite (or Smite we sometimes say, or "characteristic tetrahedron"), also same as a K-module or 1/120th of a 7.5 volumed rhombic triacontahedron.
D.M.Y. Sommerville's 1923 paper, Space-filling Tetrahedra narrows it down to the three Fuller yaks about: Mite, Bite and Rite, plus this fourth one.
This fourth one, a quarter Rite, is believed by Sommerville to round out the complete list of Euclidean space-fillers.
Whereas the Mite will assemble the Bite and the Rite, this final tetrahedral space-filler is not filled by the Mite.