Objective. We introduce a novel, phase-based, functional connectivity descriptor that encapsulates not only the synchronization strength between distinct brain regions, but also the time-lag between the involved neural oscillations. The new estimator employs complex-valued measurements and results in a brain network sketch that lives on the smooth manifold of Hermitian Positive Definite (HPD) matrices. Approach: Leveraging the HPD property of the proposed descriptor, we adapt a recently introduced dimensionality reduction methodology that is based on Riemannian Geometry and discriminatively detects the recording sites which best reflect the differences in network organization between contrasting recording conditions in order to overcome the problem of high-dimensionality, usually encountered in the connectivity patterns derived from multisite encephalographic recordings. Main results. The proposed framework is validated using an EEG dataset that refers to the challenging problem of differentiating between attentive and passive visual responses. We provide evidence that the reduced connectivity representation facilitates high classification performance and caters for neuroscientific explorations. Significance. Our paper is the very first that introduces an advanced connectivity descriptor that can take advantage of Riemannian geometry tools. The proposed descriptor, that inherently and simultaneously captures both the strength and the corresponding time-lag of the phase synchronization, is the first phase-based descriptor tailored to leverage the benefits of Riemannian geometry.