Many of us are familiar with the connection between spurious correlation and its relationship to cointegration. Granger explains in his seminal 1974 paper "Spurious Regressions in Econometrics" how hypothesis testing on two time series that are I(1) can lead to significance tests that are very biased according to experimental results (resulting in far more "significant" correlations than there should be). Later, Philips (1986) showed with functional limit theory exactly why these I(1) time series led to biased results: because in fact there is no convergent asyptotic behavior for the correlation coefficient for these series (among other relevant statistics).
At this point, there seems to be a gap in my understanding. There seems to be dozens of articles online confidently proclaiming that you can "get around" the spurious correlation problem if the two time series in question are not only I(1), but also cointegrated. The explanation seems to be that since the error term is I(0), the usual assumptions of hypothesis testing for correlation are met. Of course, this makes intuitive sense, since the original assumption for the correlation significance test is that the residuals are iid normal. But saying the residuals are I(0) is not the same as saying they are iid normal.
I am wondering if there is a reference that explains this relationship in more detail? Who was it that showed that regular hypothesis testing is valid when the two series are cointegrated? And is this an "if and only if", or are there other circumstances other than co-integration under which regular OLS correlation testing is still valid?