The fracture of vascular tissue, and load-bearing soft tissue in general, is relevant to various biomechanical and clinical applications, from the study of traumatic injury and disease to the design of medical devices and the optimisation of patient treatment outcomes. The fundamental mechanisms associated with the inception and development of damage, leading to tissue failure, have yet to be wholly understood. We present the novel coupling of a microstructurally motivated continuum damage model that incorporates the time-dependent interfibrillar failure of the collagenous matrix with an embedded phenomenological representation of the fracture surface. Tissue separation is therefore accounted for through the integration of the cohesive crack concept within the partition of unity finite element method. A transversely isotropic cohesive potential per unit undeformed area is introduced that comprises a rate-dependent evolution of damage and accounts for mixed-mode failure. Importantly, a novel crack initialisation procedure is detailed that identifies the occurrence of localised deformation in the continuum material and the orientation of the inserted discontinuity. Proof of principle is demonstrated by the application of the computational framework to two representative numerical simulations, illustrating the robustness and versatility of the formulation.