  The distribution of Coxeter descents and block number over the set of fully
commutative elements in the hyperoctahedral group $B_n$, $\FC(B_n)$, is studied
in this paper. We prove that the associated Chow quasi-symmetric generating
function is equal to a non-negative sum of products of two Schur functions. The
proof involves a decomposition of $\FC(B_n)$ into a disjoint union of two-sided
Barbash-Vogan combinatorial cells, a type $B$ extension of Rubey's descent
preserving involution on $321$-avoiding permutations and a detailed study of
the intersection of $\FC(B_n)$ with $S_n$-cosets which yields a new
decomposition of $\FC(B_n)$ into disjoint subsets called fibers. We also
compare two different type $B$ Schur-positivity notions, arising from works of
Chow and Poirier
