The connection between the random-phase approximation and the ring-coupled-cluster-doubles method bridges the gap between density-functional and wave-function theories and the importance of the random-phase approximation lies in both its broad applicability and this linking role in electronic-structure theory. In this contribution, we present an explicitly correlated approach to the random-phase approximation, based on the direct ring-coupled-cluster-doubles ansatz, which overcomes the problem of slow basis-set convergence, inherent to the random-phase approximation. Benchmark results for a test set of 106 molecules and a selection of 10 organic complexes from the S22 test set demonstrate that convergence to within 99% of the basis-set limit is reached for triple-zeta basis sets for atomisation energies, while quadruple-zeta basis sets are required for interaction energies. Corrections due to single excitations into the complementary auxiliary space reduce the basis-set incompleteness error by one order of magnitude, while contributions due to the coupling of conventional and geminal amplitudes are in general negligible. We find that a non-iterative explicitly correlated correction to first order in perturbation theory exhibits the best ratio of accuracy to computational cost.