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Inline algorithms have been proposed for coupling Monte Carlo neutron transport solvers with several other physics, such as xenon and iodine densities and thermal hydraulics. This paper proposes a new inline algorithm that can be applied to burnup calculations. The algorithm is a modification of the predictor-corrector method, where the corrector-step nuclide densities are converged simultaneously with the fission source. This could, in principle, obviate the need for two full neutronics solutions per time-step while still allowing the accuracy of predictor-corrector methods with improved stability. This paper describes the algorithm and demonstrates its stability properties through a Fourier analysis. Although not unconditionally stable, judicious use of batching and relaxation are shown to greatly improve the algorithm’s stability properties in realistic systems.