I wonder if it has something to do with Steam interacting with Windows 10. I have three machines with Steam installed, and Gal Civ III installed, one on Windows 7, one on Windows 8.1, one on Windows 10 Preview. In addition, I have two other machines with Steam installed, but without Gal Civ III.
The dates of the files that keep replicating are from 4/28 to 5/12, if that helps nail down the issue.
Thanks for all the responses here, helps to know I'm not nuts.