  We define a new family of noncommutative Bell polynomials in the algebra of
free quasi-symmetric functions and relate it to the dual immaculate basis of
quasi-symmetric functions. We obtain noncommutative versions of Grinberg's
results [Canad. J. Math. 69 (2017), 21--53], and interpret them in terms of the
tridendriform structure of WQSym. We then present a variant of Rey's self-dual
Hopf algebra of set partitions [FPSAC'07, Tianjin] adapted to our
noncommutative Bell polynomials and give a complete description of the Bell
equivalence classes as linear extensions of explicit posets.
