\textit{G. Rauzy} [Bull. Soc. Math. Fr. 110, 147--178 (1982; Zbl 0522.10032)] associated a fractal tiling of \(\mathbb R^d\) to a primitive unimodular Pisot substitution. Here, it is shown that the resulting fractal is the closure of its interior, and that a natural decomposition of the Rauzy fractal is a self-affine multitiling. This gives a construction of many aperiodic self-affine tilings. The sets arising in the above decomposition are conjectured to be measure-wise disjoint. A different proof of a theorem by \textit{P. Arnoux} and \textit{S. Ito} [Bull. Belg. Math. Soc. - Simon Stevin 8, 181--207 (2001; Zbl 1007.37001)] concerning a special case of this conjecture is presented.