  A natural extension of bipartite graphs are $d$-partite clutters, where $d
\geq 2$ is an integer. For a poset $P$, Ene, Herzog and Mohammadi introduced
the $d$-partite clutter $\mathcal{C}_{P,d}$ of multichains of length $d$ in
$P$, showing that it is Cohen-Macaulay. We prove that the cover ideal of
$\mathcal{C}_{P,d}$ admits an $x_i$-splitting, determining a recursive formula
for its Betti numbers and generalizing a result of Francisco, H\`a and Van Tuyl
on the cover ideal of Cohen-Macaulay bipartite graphs. Moreover we prove a
Betti splitting result for the Alexander dual of a Cohen-Macaulay simplicial
complex.
