  All edge-to-edge tilings of the sphere by congruent regular triangles and
congruent rhombi are classified as: (1) a $1$-parameter family of protosets
each admitting a unique $(2a^3,3a^4)$-tiling like a triangular prism; (2) a
$1$-parameter family of protosets each admitting 2 different
$(8a^3,6a^4)$-tilings like a cuboctahedron and a triangular orthobicupola
respectively; (3) a sequence of protosets each admitting a unique
$(2a^3,(6n-3)a^4)$-tiling like a generalized anti-triangular prism for each
$n\ge3$; (4) 26 sporadic protosets, among which nineteen admit a unique tiling,
one admits 3 different tilings, one admits 5 different tilings, three admit 2
different tilings, two admit too many tilings to count. The moduli of
parameterized tilings and all geometric data are provided.
