Companies often specify the intended behaviour of their business processes in a process model. Conformance checking techniques allow us to assess to what degree such process models and corresponding process execution data correspond to one another. In recent years, alignments have proven extremely useful for calculating conformance checking statistics. Existing techniques to compute alignments have been developed to be used in an offline, a posteriori setting. However, we are often interested in observing deviations at the moment they occur, rather than days, weeks or even months later. Hence, we need techniques that enable us to perform conformance checking in an online setting. In this paper, we present a novel approach to incrementally compute prefix-alignments, paving the way for real-time online conformance checking. Our experiments show that the reuse of previously computed prefix-alignments enhances memory efficiency, whilst preserving prefix-alignment optimality. Moreover, we show that, in case of computing approximate prefix-alignments, there is a clear trade-off between memory efficiency and approximation error.