The clique problem with multiple-choice constraints (CPMC) represents a very common substructure in many real-world applications, for example scheduling problems with precedence constraints. It consists in finding a clique in a graph whose nodes are partitioned into subsets, such that exactly one node from each subset is chosen. Even though we can show that (CPMC) is NP-complete, we will be able to identify and characterize a new interesting subclass which is solvable in polynomial time. It arises whenever there are no circular dependencies between the subsets in the partition when it comes to the compatibilities of their elements. We will be able to show that this subclass is equivalent to the clique problem on a perfect graph with certain additional structure. This finding will allows us to give a full convex-hull description. Although the convex hull can have exponentially-many facets, we can state an efficient separation algorithm. In an application from underground and railway timetabling, we show that these results lead to significant reductions in computation time.