Catalyzes the rearrangement of 1-deoxy-D-xylulose 5-phosphate (DXP) to produce the thiazole phosphate moiety of thiamine. Sulfur is provided by the thiocarboxylate moiety of the carrier protein ThiS. In vitro, sulfur can be provided by H(2)S. 1-deoxy-D-xylulose 5-phosphate + 2-iminoacetate + [sulfur-carrier protein ThiS]-C-terminal Gly-NH-CH2-C(O)SH = 2-[(2R,5Z)-2-carboxy-4-methylthiazol-5(2H)-ylidene]ethyl phosphate + [sulfur-carrier protein ThiS]-C-terminal Gly-Gly + 2 H(+) + 2 H2O Cofactor biosynthesis; thiamine diphosphate biosynthesis. Homotetramer. Forms heterodimers with either ThiH or ThiS. Belongs to the ThiG family.