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Cosmic reionization was driven by the imbalance between early sources and sinks of ionizing radiation, both of which were dominated by small-scale structure and are thus usually treated in cosmological reionization simulations by subgrid modelling. The recombination rate of intergalactic hydrogen is customarily boosted by a subgrid clumping factor, 〈n2〉/〈n〉2, which corrects for unresolved fluctuations in gas density n on scales below the grid-spacing of coarse-grained simulations. We investigate in detail the impact of this inhomogeneous subgrid clumping on reionization and its observables, as follows: (1) Previous attempts generally underestimated the clumping factor because of insufficient mass resolution. We perform a high-resolution N-body simulation that resolves haloes down to the pre-reionization Jeans mass to derive the time-dependent, spatially varying local clumping factor and a fitting formula for its correlation with local overdensity. (2) We then perform a large-scale N-body and radiative transfer simulation that accounts for this inhomogeneous subgrid clumping by applying this clumping factor-overdensity correlation. Boosting recombination significantly slows the expansion of ionized regions, which delays completion of reionization and suppresses 21 cm power spectra on large scales in the later stages of reionization. (3) We also consider a simplified prescription in which the globally averaged, time-evolving clumping factor from the same high-resolution N-body simulation is applied uniformly to all cells in the reionization simulation, instead. Observables computed with this model agree fairly well with those from the inhomogeneous clumping model, e.g. predicting 21 cm power spectra to within 20 per cent error, suggesting it may be a useful approximation.