  We construct bases of quasi-symmetric functions whose product rule is given
by the shuffle of binary words, as for multiple zeta values in their integral
representations, and then extend the construction to the algebra of free
quasi-symmetric functions colored by positive integers. As a consequence, we
show that the fractions introduced in [Guo and Xie, Ramanujan Jour. 25 (2011)
307-317] provide a realization of this algebra by rational moulds extending
that of free quasi-symmetric functions given in [Chapoton et al., Int. Math.
Res. Not. IMRN 2008, no. 9, Art. ID rnn018].
