The answer is no because you may have injected this phenomenon as a result of transforming the data in an unwarranted fashion ... see the Slutsky Effect where a linear (weighted ) combinations of i.i.d. values leads to a series with auto-correlative structure . Slutsky http://www-history.mcs.st-andrews.ac.uk/Biographies/Slutsky.html Effect ... Unnecesaary differencing can INJECT variability. Consider the variance of a random process that is differenced OR unnecessarily filtered http://mathworld.wolfram.com/Slutzky-YuleEffect.html
Non-stationarity is a symptom with possibly many causes. One cause is a shift in the mean at one or more points in time. Another possible cause is a change in parameters at one or more points in time. Another cause is a deterministic change in error variance at one or more points in time. Prof. Spyros Makridakis wrote an article http://www.insead.edu/facultyresearch/research/doc.cfm?did=46900 of the danger of using differencing to render a series stationary.
When (and why) should you take the log of a distribution (of numbers)? discusses when you should take a power transform i.e. to decouple the relationship between the Expected Value and the variance of the model's residuals.
You may be injecting structure via unwarranted transformations ( differencing is a transformation) .
Simply adjusting for a contemporaneous series (inflation) may be incorrect as the Y variable may be impacted by changes in the X variable or lags of the X variable.
This is why we build SARMAX models https://autobox.com/pdfs/SARMAX.pdf.
Why don't you post your original data in a csv format and I and others may
be able to help .
EDITED AFTER RECEIPT OF DATA:
I took your 132 monthly values into AUTOBOX ( a piece of software that I have helped to develop ) and automatically developed a useful model . It has a number of advanced features that can be helpful.
Here is the data which clearly suggests that as the series gets higher the variability increases. An even "truer" statement is that the variance changes at one point in time (around period 54) and not pervasively suggesting that a Weighted least Squares would be more appropriate than a Log Transform . This will be found via the TSAY test described here https://onlinelibrary.wiley.com/doi/abs/10.1002/for.3980070102 with an excerpt here
The TSAY test shown here led to a first difference model (nearly second differences as suggested by the ar coefficients nearly summing to 1.0 ) here with 9 pulses/shocks and a positive level shift (intercept change) at period 68.
The model in more detail is here and here
The Actual , Fit and Forecast graph is here with MOnte-Carlo generated simulations leading to these forecasts and limits
The role of statistics is to separate the data into signal and noise thus the litmus test is "did the equation generate a suitable noise process" . I would say a loud "Yes" .
Here is the plot of the model's residuals with this acf
In summary a useful model requires that the data be treated for non-constant variance by employing Weighted Least Squares effectively discounting the values 54-132 . The arima model is (2,1,0)(0,0,0)12 with a constant and 1 level shift along with 9 pulses.
It can help to see a segment of the augmented data matrix with the pulses and level shift where the columns represent the latent deterministic structure that was "scraped" from the data .
Hope this helps you and the list better ( partially ) understand the extraction of signal from data. No seasonality is detected with the data given .