  This is an extended abstract for a talk given at the mini-workshop
"Cohomology rings and fundamental groups of hyperplane arrangements, wonderful
compactifications, and real toric varieties", held in Oberwolfach, September
30-October 6, 2012. We describe a formula (obtained in joint work with Alvise
Trevisan) for computing the Betti numbers of real toric manifolds, based on a
study of cohomology with coefficients in rank one local systems for various
polyhedral products. Applying this formula, we recover Henderson's computation
of the Betti numbers of real Hessenberg varieties. We conclude with a
discussion of the formality question in this context.
