Underwater attachment is a significant challenge, for which we have no good general solutions in our technology. Yet, a number of biological organisms have evolved solutions to this problem. One intriguing approach to underwater attachment is that of the marine bivalve mussel Anomia simplex that uses a biomineralized byssus to permanently anchor itself to substrates. The byssus has a highly complex hierarchical structure and contains over 90 wt% CaCO3. The byssus features a complex set of porosities, presumed to be highly important for the function of the attachment system. The pore space is the main focus of the present work. We characterize the three dimensional distribution of pore spaces in the byssus using micro-computed tomography (µCT) through a combination of in house CT and high-resolution synchrotron CT. The pore structures are observed to fall into distinct categories in various parts of the byssus. We discuss the branching of one set of pores that reach the byssus substrate interface in particular. They form a network reaching the byssus surface that we now visualize in three dimensions.