This is a wild correlation
There's ENSO, which is a standing-wave oscillation and there's the Madden-Julian Oscillation (MJO), which is a traveling-wave that propagates eastward a bit like an erratic sequence of [solitons](https://en.wikipedia.org/wiki/Soliton).
Found a correlation between the two by analyzing the high-resolution daily Southern Oscillation Index (SOI) of ENSO and then duplicated the amplitude and phase of the MJO by shifting it forward by ~21 days. IOW, the SOI leads the MJO by 21 days.
No one in the climate research literature has reported this as far as I can tell, even though it is blatantly obvious.
The mechanism is straightforward to explain -- transitions in the ENSO forcing (including El Nino/La Nina transitions) are the trigger for the MJO traveling wave, and it likely takes at least several days for the traveling wave to fully propagate to the extent it can be measured. Perhaps think in terms of a visual lightning strike and then hearing the thunder, propagating at 7 seconds per mile, but in this case the MJO traveling wave is only moving at ~5 meters per second.
This paper explains how a standing wave and traveling wave can be isolated
The left column is the combined waveform and the two on the right are the SW & TW decomposition.