We investigate the role of microstructural bridging on the fracture toughness of composite materials. To achieve this, a new computational framework is presented that integrates phase field fracture and cohesive zone models to simulate fibre breakage, matrix cracking and fibre-matrix debonding. The composite microstructure is represented by an embedded cell at the vicinity of the crack tip, whilst the rest of the sample is modelled as an anisotropic elastic solid. The model is first validated against experimental data of transverse matrix cracking from single-notched three-point bending tests. Then, the model is extended to predict the influence of grain bridging, brick-and-mortar microstructure and 3D fibre bridging on crack growth resistance. The results show that these microstructures are very efficient in enhancing the fracture toughness via fibre-matrix debonding, fibre breakage and crack deflection. In particular, the 3D fibre bridging effect can increase the energy dissipated at failure by more than three orders of magnitude, relative to that of the bulk matrix; well in excess of the predictions obtained from the rule of mixtures. These results shed light on microscopic bridging mechanisms and provide a virtual tool for developing high fracture toughness composites.