We use dual porosity microfluidics and fluorescence microscopy to investigate immiscible imbibition in the pore networks formed in fractured rocks, and to identify emergent pore-scale events that arise as a result of the interplay between advection-dominant flow in fractures (F) and capillary-driven matrix imbibition (M). The dimensionless ratio between the two time scales T = tF/tM defines the various displacement patterns: fracture-dominant advective invasion at low Τ-values leaves a higher residual non-wetting phase saturation; compact invasion is observed at intermediate T-values, and fractures act as capillary barriers during matrix-dominant capillary imbibition at high Τ-values. Experiments and analyses show effective capillary-driven corner flow during immiscible imbibition; in particular, corner flow imbibition displaces non-wetting fluids that were initially trapped in the matrix during fast advective invasion. In contrast to wetting fluid invasion and imbibition, injected non-wetting fluids invade and flow along fractures as soon as the capillary pressure reaches the fracture entry pressure, and there is no matrix invasion and drainage. The capillary pressure versus saturation curve for the fractured rock mass assumes that fractures and matrix blocks share the same capillary pressure at equilibrium; then, the combined pressure-saturation response is a function of their relative contributions to the total porosity. In the absence of gouge or precipitates, fractures determine the entry pressure while the matrix controls storativity.