Scheduling is probably the most obvious example of a real problem. You’ve got a space that has many different ways of being bounded and need to efficiently search that space.

Physical design systems (modeling possible boat hulls, say—not the physics itself but the interaction of numerous variables) and resource allocation (drafting a baseball team, say) seem like other likely areas, though I’ve got no idea if constraint tools are the best option or whether they’re even used.

Broadly: if a multivariate problem can’t be modeled, a constraint-based search strategy is the(/one) next-best tool.