It takes several thousand years for the deep-ocean concentration of natural radiocarbon to come to equilibrium with surface fluxes, making it computationally too expensive to routinely simulate it with moderate- to high-resolution ocean models. We present an implicit solver for computing prebomb Δ¹⁴C that requires the equivalent of only a few tens of model years to reach equilibrium. The solver uses a Newton–Krylov algorithm with a preconditioner based on a coarse-grained annually-averaged tracer-transport operator. Coarse-graining provides a general approach for developing preconditioners for models of increasing resolution. We implemented and tested the solver for the ocean component of the Community Earth System Model (CESM) with a nominal horizontal resolution of 1° ×1° and with 60 vertical levels. Simulated Δ¹⁴C values are in good agreement with observations at the surface and in the North Atlantic, but the deep North Pacific simulated values show a substantial bias, with prebomb radiocarbon Δ¹⁴C values translating to ages that are twice the observationally based estimate. This bias is substantially larger than published simulations obtained with coarser resolution models, suggesting that increasing model resolution does not automatically improve the fidelity of the deep ocean ventilation processes. We therefore recommend that natural Δ¹⁴C be used as a deep-ocean ventilation metric for critically evaluating deep ocean circulation.