The electroencephalogram (EEG) recorded during magnetic resonance imaging (MRI) inside the scanner is obstructed by the MRI gradient artefact (MGA) originating from the electromagnetic interference of the MRI with the sensitive measurement of electrical scalp potentials. Post-processing algorithms based on average artefact subtraction (AAS) have proven to be efficient in removing the MGA. However, the residual MGA after AAS still limits the quality and usable bandwidth of the EEG data despite further reduction through re-sampling, principal component analysis (PCA), and regressive filtering. We recently demonstrated that the residual MGA can largely be avoided by means of hardware synchronisation. Here we present a new software synchronisation method, which substitutes hardware synchronisation and facilitates the removal of motion artefacts by PCA. The effectiveness of the retrospective synchronisation algorithm (Resync) is demonstrated by comparison to the aforementioned techniques. For this purpose we also developed a method for simulating the MGA and we propose new concepts for quantifying and comparing the performance of post-processing algorithms for EEG-MRI data. Results indicate that the benefits of (retrospective) synchronisation and PCA depend largely on the relative contribution of timing errors and motion artefacts to the residual MGA as well as the frequency range of interest.