Coated-fiber composites are of continuing technological interest. The purposes of the coating may include protecting the fibers from elevated-temperature oxidation, softening the fiber-matrix interfacial stress concentration, and increasing the work of fracture or the damping of the composite. However, thin brittle coatings on circular-section substrates subjected to uniaxial extension have been observed to have a regular fracture pattern due to buckling of the coating. This buckling-induced fracture phenomenon is due to coating circumferential compression induced by the larger Poisson's ratio of the substrate relative to the fiber and was recently analyzed. Also the buckling of the coating in a coated-fiber composite was analyzed very recently. The present work considers the case of a composite containing coated transversely isotropic fibers so as to be applicable to the technologically important case of graphite fibers.