Real-time time-dependent density functional theory (RT-TDDFT) has become a widely used first-principles method for studying excited-state and nonequilibrium electron dynamics. Hybrid exchange-correlation (XC) functionals, which incorporate a fraction of nonlocal exact exchange, are particularly important for accurately describing processes such as charge transfer and hot-carrier dynamics. However, the high computational cost which accompanies exact exchange, as required in hybrid XC functionals, has largely restricted RT-TDDFT simulations of condensed matter systems to using semi-local XC functionals. In this work, we present an all-electron implementation of exact exchange for RT-TDDFT based on numeric atom-centered orbitals, extending an efficient ground-state resolution-of-identity (RI) framework to the complex, time-dependent Hamiltonians required for real-time propagation. We validate the implementation against an established RI approach, for an isolated water molecule, finding exact agreement for both valence and core (K-edge) excitations. We also demonstrate the method on two extended periodic systems, crystalline silicon and silicon carbide. For silicon, the HSE06 spectrum reproduces previously reported real-time TDDFT results and agrees with experiment to within approximately 0.2 eV, consistent with plane-wave pseudopotential implementations. For silicon carbide, we compute valence and X-ray absorption spectra at the silicon L-edge and carbon K-edge. These results establish an efficient, all-electron route to hybrid-XC functional RT-TDDFT for both molecular and condensed matter systems.