Anomalous transport is ubiquitous in a wide range of disordered systems, notably in fractured porous formations. We quantitatively identify the structural controls on anomalous tracer transport in a model of a real fractured geological formation that was mapped in an outcrop. The transport, determined by a continuum scale mathematical model, is characterized by breakthrough curves (BTCs) that document anomalous (or “non-Fickian”) transport, which is accounted for by a power-law distribution of local transition times ψ(t) within the framework of a continuous time random walk (CTRW). We show that the determination of ψ(t) is related to fractures aligned approximately with the macroscopic direction of flow. We establish the dominant role of fracture alignment, and assess the statistics of these fractures by determining a concentration-visitation weighted residence time histogram. We then convert the histogram to a probability density function (pdf) that coincides with the CTRW ψ(t) and hence anomalous transport. We show that the permeability of the geological formation hosting the fracture network has a limited effect on the anomalous nature of the transport; rather, it is the fractures transverse to the flow direction that play the major role in forming the long BTC tail associated with anomalous transport. This is a remarkable result, given the complexity of the flow field statistics as captured by concentration transitions.