I am currently thinking that perhaps the ENSO and QBO are being forced by the same underlying factors, and what we are seeing WRT the difference in the two responses results from the characteristics of the medium that generates the response. So that the QBO, characterized by a low density medium (thin air), is able to respond quickly and thus has a very high characteristic frequency. But the ENSO, characterized by the sluggish response of water, must have a much lower characteristic frequency.
An interesting hypothesis is that the draconic lunar tide of duration 27.2122 days may be the common underlying significant driver. Unfortunately, the QBO and ENSO data are sampled at only a monthly rate, so we can’t do much to pull out the signal intact from our data ... Or can we?
What’s intriguing is that the driving force isn’t at this monthly level anyways, but likely is the result of a beat of the monthly tidal signal with the yearly signal. It is expected that strong tidal forces will interact with seasonal behavior in such a situation and that we should be able to see the effects of the oscillating tidal signal where it constructively interferes during specific times of the year. For example, a strong tidal force during the hottest part of the year, or an interaction of the lunar signal with the solar tide (a precisely 6 month period) can pull out a constructively interfering signal.
To analyze the effect, we need to find the tidal frequency and un-alias the signal by multiples of 2$\pi$
So that the draconic frequency of 2$\pi$/(27.212/365.25) = 84.33 rads/year becomes 2.65 rads/year after removing 13 x 2$\pi$ worth of folded signal. This then has an apparent period of 2.368 years. Compare that to the nominal 2.33 year period of QBO, and the strong spectral component of ENSO at this frequency (see comment #18), and the figure below.
The period of 2.368 years is just to the left of the peaks -- it should be lined up at 0.22 on the x-axis. That is agonizingly close, but over the course of 100 years, the difference in propagated phase error between 2.368 and 2.333 is about one-half the average cycle. But then again, 2.368 could be the actual QBO period since we only have ~60 years of data, and so could easily be off by a ¼ of a cycle. It could be that the 7-year syncing of 2.33 x 3 is enough to push it towards that value.
Next, consider the draconic period in its bi-monthly form against that of the Chandler wobble period of 432.5 days ( for a constant seasonal signal ). This comes out to 2.3681/2 years when un-aliased or 1.184 years = 432.47 days . Consider that the beat frequency of the Chandler wobble is 1/(1-1/1.184) = 6.43 years, which shows up in the ENSO signal. Is this a coincidence or a related forcing?
The other strong QBO-like signal is shown in the figure below labeled “Folded Mean Lunation”.
The unaliased synodic lunar tide forcing comes out to be 2$\pi$/(29.531/365.25) = 77.71. After removing 12*2$\pi$ worth of aliased signal, this comes out to 2.314 rads/year or a period of 2.71 years, which is close to the secondary signal shown. What’s more is that the synodic frequency is not steady over time and so will split into sidebands as the lunation period will cycle between maximum and minimum values. This can explain why there are satellite peaks as these will again get aliased and form a beat pattern with specific seasons.
And finally consider the fact that these two tidal signals -- draconic and synodic -- appear in both ENSO and QBO. I found the synodic signal in the QBO using Eureqa where it actually pulled the high frequency signal out of the data. Read this post from over a year ago, when I first observed the aliasing:
Eureqa picked out (with abslutely none of my help) the frequency components
77.72 rads/year → 29.53 days = synodic month
153 rads/year → 15 days = 1/2 synodic month
72.73 rads/year → aliased 2.355 years, close to aliased period of 2.368 years for draconic month
(72.73 - 12 2$\pi$ = -2.67 rads/yr)
so the numbers actually work a bit better for QBO than for ENSO, but the bottomline fact that they have this synodic and draconic lunar month commonality is striking.
As a caveat, these numbers really need to be spot on to make sense because tidal frequencies are so well-characterized. Anything that is off by a little bit will propagate as an error over the 100+ year range we are dealing with. They are tantalizing close though.
 MASAKI, Yoshimitsu. "Expected Seasonal Excitations of Earth Rotation by Unmodeled Geophysical Fluids." Bulletin of the Geographical Survey Institute 54 (2007): 2.