Rotating spiral waves with a central core composed of phase-randomized oscillators can arise in reaction-diffusion systems if some of the chemical components involved are diffusion-free. This peculiar phenomenon is demonstrated for a paradigmatic three-component reaction-diffusion model. The origin of this anomalous spiral dynamics is the effective nonlocality in coupling, whose effect is stronger for weaker coupling. There exists a critical coupling strength which is estimated from a simple argument. Detailed mathematical and numerical analyses are carried out in the extreme case of weak coupling for which the phase reduction method is applicable. Under the assumption that the mean-field pattern keeps rotating steadily as a result of a statistical cancellation of the incoherence, we derive a functional self-consistency equation to be satisfied by this space-time dependent quantity. Its solution and the resulting effective frequencies of the individual oscillators are found to agree excellently with the numerical simulation.