The limiting efficiency for series-connected multijunction solar cells is usually calculated from the assumption that the individual junctions are optically isolated. Here, we develop an analytical formalism to predict efficiencies attainable in the presence of luminescent coupling, i.e., if the individual junctions in a series-connected multijunction stack are coupled optically, so that luminescence from one junction can be absorbed by the lower bandgap junction below. The formalism deals with nonradiative recombination through the definition of the luminescence extraction efficiency. Using our general formalism, we find that the limiting efficiency of a tandem cell becomes much less dependent on exact bandgap combination when luminescent coupling is considered and proceed to consider two technologically important examples of current-mismatched tandem solar cells. We find that a series-connected GaAs on a silicon tandem cell can be more efficient than the underlying silicon cell alone, if the luminescence extraction efficiency of the GaAs junction is sufficient. An analysis of luminescent coupling in a perovskite on a silicon tandem cell shows that the efficiency penalty for a perovskite bandgap below the optimum value can be mitigated if the luminescence extraction efficiency is high. We suggest that material quality and stability might be more important considerations for perovskite on silicon tandems than engineering the bandgap to achieve precise current matching.