  This paper investigates the rational Betti numbers of real toric manifolds
associated with chordal nestohedra. We consider the poset topology of a
specific poset induced from a chordal building set, and show its
EL-shellability. Based on this, we present an explicit description using
alternating $\mathcal{B}$-permutations for a chordal building set
$\mathcal{B}$, transforming the computing Betti numbers into a counting
problem. This approach allows us to compute the $a$-number of a finite simple
graph through permutation counting when the graph is chordal. In addition, we
provide detailed computations for specific cases such as real Hochschild
varieties corresponding to Hochschild polytopes.
