Catalyzes the post-translational addition of a tyrosine to the C-terminal end of detyrosinated alpha-tubulin. ATP + C-terminal L-alpha-aminoacyl-L-glutamyl-L-glutamyl-[tubulin] + L-tyrosine = ADP + C-terminal L-alpha-aminoacyl-L-glutamyl-L-glutamyl-L-tyrosyl-[tubulin] + H(+) + phosphate Monomer. Belongs to the tubulin--tyrosine ligase family.