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A similar error was solved in Excel 2010.
Kindly refer the following link and see if that helps you.
https://support.microsoft.com/en-us/kb/2597034#/en-us/kb/2597034
Thank you for the tip, but this did not solve the problem. In all fairness, I didn't install the hot fix since it is for Excel 2010 and I'm running 2013, but the registry edit itself did not fix the problem.
A little more detail as I just noticed something: when adding the new pivot from the existing data set I am selecting the column headers and not explicitly where the data starts and ends. This normally shouldn't be an issue, but since it's recognizing using the same data set as I have previously for other pivots, it's also using the setting of not saving the data source with the pivot, so when add an initial item to the new pivot I get the 'The PivotTable was saved without the underlying data. Use the Refresh Data command to update the report' pop-up message. I refresh, then add items to ROWS, COLUMNS, VALUES, etc. At this point when I try to save I get the originally listed error message and I am unable to save.
I just tried by adding a pivot using the explicit data range, which adds the pivot and has the 'Save source data with file' enabled and I am able to save the file with no issues. If I then change the data range for the new pivot to reference just the columns, I get the same 'Refresh' pop-up, but the 'Save data source' is still enabled and I am still able to save the file.
I hope this is clear and makes sense.