Solid-state electronic spins are extensively studied in quantum information science, as their large magnetic moments offer fast operations for computing1 and communication2-4, and high sensitivity for sensing5. However, electronic spins are more sensitive to magnetic noise, but engineering of their spectroscopic properties, for example, using clock transitions and isotopic engineering, can yield remarkable spin coherence times, as for electronic spins in GaAs6, donors in silicon7-11 and vacancy centres in diamond12,13. Here we demonstrate simultaneously induced clock transitions for both microwave and optical domains in an isotopically purified 171Yb3+:Y2SiO5 crystal, reaching coherence times of greater than 100 μs and 1 ms in the optical and microwave domains, respectively. This effect is due to the highly anisotropic hyperfine interaction, which makes each electronic-nuclear state an entangled Bell state. Our results underline the potential of 171Yb3+:Y2SiO5 for quantum processing applications relying on both optical and spin manipulation, such as optical quantum memories4,14, microwave-to-optical quantum transducers15,16, and single-spin detection17, while they should also be observable in a range of different materials with anisotropic hyperfine interactions.