Extreme anisotropy in some polaritonic materials enables light propagation with a hyperbolic dispersion, leading to enhanced light-matter interactions and directional transport. However, these features are typically associated with large momenta that make them sensitive to loss and poorly accessible from far-field, being bound to the material interface or volume-confined in thin films. Here, we demonstrate a new form of directional polaritons, leaky in nature and featuring lenticular dispersion contours that are neither elliptical nor hyperbolic. We show that these interface modes are strongly hybridized with propagating bulk states, sustaining directional, long-range, sub-diffractive propagation at the interface. We observe these features using polariton spectroscopy, far-field probing and near-field imaging, revealing their peculiar dispersion, and – despite their leaky nature – long modal lifetime. Our leaky polaritons (LPs) nontrivially merge sub-diffractive polaritonics with diffractive photonics onto a unified platform, unveiling opportunities that stem from the interplay of extreme anisotropic responses and radiation leakage.