I created another variant of the chart in #10. Besides the biennial forcing function on the RHS of the model, I added annual and biannual forcing factors to give the model fine structure. This doesn't improve the correlation coefficient significantly but it indicates where the details in the signal may be coming from.
The fit is qualitatively good and it appears on the verge of matching exactly, except for areas highlighted in yellow. These appear to flip as mirror images around the zero level, perhaps indicating a phase reversal on a zero crossing. If the forcing indeed is a biennial function, the phase on an even and odd year is metastable, as neither is physically preferred.
Post year 2000, the phase reverses completely, and the forcing waveform needs to be shifted to get the model in synch with the tidal data:
This is uncharted territory AFAIAC and so it will be interesting what it leads to.