trend stationary with external regressors Suppose I have two trend - stationary time series with strong correlation. 
In the case where there are no regressors, if a time series is trend-stationary, it becomes stationary by subtracting a deterministic trend. Meaning, I fit a line to the data $Y=f(t)$ and subtract that line from the data. Then I fit the residuals with Arima. 
In the case where there is an independent regressor... do I subtract the best fit line from both $Y = f_1(t),\, X = f_2(t)$, or do I subtract lines : $Y = f_1(X, t), \, X = f_2(Y, t)$, where $X$ is the regressor, $Y$ is the dependent time series variable, and $t$ is time. 
I think the answer is to subtract a best fit line which only considers time, because stationary means the covariance and expectations are constant functions of time... but I am looking for a second opinion from you geniuses. 
I worry because I don't want to lose the correlation between regressor and dependent variable by subtracting best fit trend lines from the data. 
 A: It appears to me you are struggling with presumed deterministic time trends and possible deterministic level shifts in one or more series that you are trying to relate.
Subtraction is not the answer ...filtering via pre-whitening is the GENERALLY preferred method to identify a possible useful model which can then lead to identifying time trends and/or level shifts  GIVEN the impact on Y of the candidate X. 
Take a look at two web references that I think will help you going forward.
http://www.math.cts.nthu.edu.tw/download.php?filename=569_fe0ff1a2.pdf&dir=publish&title=Ruey+S.+Tsay-Lec1 and https://newonlinecourses.science.psu.edu/stat510/lesson/9/9.1 culminating in 
 https://autobox.com/pdfs/SARMAX.pdf 
If you wish to pursue this thread , perhaps in another question actually create via simulation a test case where you have embedded certain (possibly identifiable) structures and post it to challenge the readers of this list. Simulation can be a great teacher and an even greater software evaluation tool/strategy.
