In this work we study the rate-distortion function (RDF) for lossy compression of asynchronously-sampled continuous-time (CT) wide-sense cyclostationary (WSCS) Gaussian processes with memory. As the case of synchronous sampling, i.e., when the sampling interval is commensurate with the period of the cyclostationary statistics, has already been studied, we focus on discrete-time (DT) processes obtained by asynchronous sampling, i.e., when the sampling interval is incommensurate with the period of the cyclostationary statistics of the CT WSCS source process. It is further assumed that the sampling interval is smaller than the maximal autocorrelation length of the CT source process, which implies that the DT process possesses memory. Thus, the sampled process is a DT wide-sense almost cyclostationary (WSACS) processes with memory. This problem is motivated by the fact that man-made communications signals are modelled as CT WSCS processes; hence, applications of such sampling include, e.g., compress-and-forward relaying and recording systems. The main challenge follows because, with asynchronous sampling, the DT sampled process is not information-stable, and hence the characterization of its RDF should be carried out within the information-spectrum framework instead of using conventional information-theoretic arguments. This work expands upon our previous work which addressed the special case in which the DT process is independent across time. The existence of dependence between the samples requires new tools to obtain the characterization of the RDF.