Summary: 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.