Finding a forbidden subgraph characterization of circular-arc graphs is a challenging open problem. Many partial results toward this goal have been proposed over the years, but a satisfactory answer has so far eluded us. In this paper, we suggest a new direction in this line of research. We propose a novel structural obstruction to circular-arc graphs---a blocking quadruple---and study its use in characterizing circular-arc graphs within chordal graphs. Notably, we observe that the absence of blocking quadruples unifies characterizations of various known chordal subclasses of circular-arc graphs found in the literature. To this end, we provide a forbidden induced subgraph characterization of chordal graphs without blocking quadruples and show that the absence of blocking quadruples exactly characterizes chordal circular-arc graphs of independence number 4 or less. Our proof uses an interesting geometric approach, constructing a circular-arc representation by traversing around a carefully chosen clique tre...