The shells of rhynchonelliform brachiopods have an outer (primary) layer of acicular calcite and an inner (secondary) layer of calcite fibres which are parallel to the shell exterior. Atomic force microscopy (AFM) reveals that these fibres are composed of large triangular nanogranules of about 600-650 nm along their long axis. The nanogranules are composites of organic and inorganic components. As the shell grows, the fibres elongate with the calcite c-axis perpendicular to the fibre axis as demonstrated by electron backscatter diffraction (EBSD). Thus, despite being a composite structure comprising granules that are themselves composites, each fibre is effectively a single crystal. The combination of AFM and EBSD reveals the details of the structure and crystallography of these fibres. This knowledge serves to identify those aspects of biological control that must be understood to enable comprehension of the biological control exerted on the construction of these exquisite biomineral structures.