Early markers of tendinopathy are related to the emergence of kink patterns that disrupt the microenvironment of tenocytes (tendon cells). Tendons pose a unique paradigm for the design of engineering soft composites, as significant plasticity and dissipation is observed at the single fibril level, which constitutes the main load bearing element. Additionally, when collagen fibrils are hierarchically assembled to form fibers and fascicles, the instability cascade deviates from the traditionally explored modes of engineering composites. This work focuses on a numerical investigation of the effect that geometrical imperfections of the fibrillar microstructure have on soft fiber reinforced composites undergoing finite deformations. Having the biological composition of tendon as a springboard, the underlying microstructure is approximated as a laminate composite, with the stiff elastoplastic fiber layer and the soft purely elastic matrix layer. %Even though the oversimplified representation of these materials as being free of geometrical variations allows analytical results, this assumption does not have its roots in biological evidence. Hence, Geometrical imperfections are introduced to encompass the natural crimp of collagen fibrils in tendon and enable obtaining numerical results in the context of computational homogenization. The theoretical framework, in which instabilities are treated as bifurcations that may be predicted through the loss of ellipticity of the governing equations, is employed to compare the behavior of geometrically perfect and imperfect models, for the latter of which the post-bifurcated response is attainable. Physical observations suggest that there are two classes of instabilities that may arise during tendon operation. These modes manifest at different length scales and are associated with distinct material properties of tendons serving different functions. Namely, a domain formation reminiscent of twinning is observed during the elastic unloading that correspond to loss of ellipticity of the composite, and micro-necking occurs during elastoplastic loading corresponding to loss of ellipticity of the fiber phase. These finding help bridge the gap to understand the evolution of crimp, the response in the toe-region, and the cell microenvironment.