The Astudillo paper asserted that ENSO was deterministic apart from the disturbance right after 1980. They did not mention a phase flip specifically.
The phase reversal idea came from consideration of this disturbance with respect to a biennial mode, with supplemental references here:
And this wavelet graph by Roundy shows how standing modes of the SST got inverted after 1980 but came back in sync around 2000:
Roundy, P.E. "On the interpretation of EOF analysis of ENSO, atmospheric Kelvin waves, and the MJO." Journal of Climate 28.3 (2015): 1148-1165.