The analysis of thin tubular objects in 3D images is a complex and challenging task. In this article, we introduce a new, nonlinear operator, called RORPO (Ranking Orientation Responses of Path Operators). Inspired by the multidirectional paradigm currently used in linear filtering for thin structure analysis, RORPO is built upon the notion of path operator from mathematical morphology. This operator, unlike the gold-standard Hessian-based operators commonly used for 3D tubular structure analysis, is discrete, non linear and non-local. From this new operator, two main tubular structure characteristics can be estimated: an intensity feature, that can be assimilated to a quantitative measure of tubularity; and a directional feature, providing a quantitative measure of the tubular structure orientation. We provide a full description of the structural and algorithmic details for computing these two features from RORPO, and we discuss computational issues. We experimentally assess RORPO by comparing with gold standard Vesselness and we show that our method performs better on both features in realistic conditions.