Living tissues, heterogeneous at the microscale, usually scatter light. Strong scattering is responsible for the whiteness of bones, teeth, and brain and is known to limit severely the performances of biomedical optical imaging. Transparency is also found within collagen-based extracellular tissues such as decalcified ivory, fish scales, or cornea. However, its physical origin is still poorly understood. Here, we unveil the presence of a gap of transparency in scattering fibrillar collagen matrices within a narrow range of concentration in the phase diagram. This precholesteric phase presents a three-dimensional (3D) orientational order biomimetic of that in natural tissues. By quantitatively studying the relation between the 3D fibrillar network and the optical and mechanical properties of the macroscopic matrices, we show that transparency results from structural partial order inhibiting light scattering, while preserving mechanical stability, stiffness, and nonlinearity. The striking similarities between synthetic and natural materials provide insights for better understanding the occurring transparency.