Growth of GaInTlAs alloys on InP(001) has been attempted by solid source molecular beam epitaxy. Thallium incorporation into Ga1-xInxAs matrices was studied as a function of substrate temperature, arsenic overpressure, matrix composition, and growth rate. At high temperatures (> 350 degreesC) thallium evaporates, whereas at intermediary temperatures (270-350 degreesC) thallium segregates into droplets on the surface. Only in the low temperature range (180-260 degreesC) can thallium be incorporated ill some conditions, leading to mirror-like surfaces. Up to 18% Tl content was incorporated into a Ca0.70In0.30As matrix and up to 40% Tl into a GaAs matrix. For these high Tl concentrations, Tl droplets are avoided and Tl incorporation is achieved only when using high arsenic pressures. However, this limits surface adatom diffusion and leads to amorphous, polycrystalline, or twinned materials. Finally, a narrow window for single-crystal growth has been found for low Tl contents (4%) using optimized growth conditions with low V/III pressure ratios and high growth rates.