As a followup to the previous post, I found a more "user-friendly" workaround. Again, not a solution, but a better workaround than others suggested (if I may be so bold as to claim). I also attached three pictures to show anyone reading this what option/checkbox I am referring to in the "Options" dialog.
Perform the following steps:
1. Upon initially opening a drawing, make sure that in the "Options" dialog (type OPTIONS in the command line), you select the "Display" tab and have "Display File Tabs" checked. If it was on to begin with, you should have noticed a tab with your drawing's name up top somewhere (location is workspace dependent). If it is off to begin with, simply check it and press APPLY, then OK. Note: There is a chance that even checking this option and pressing APPLY then OK will not work, forcing you to close AutoCAD and restart it. However, upon restarting and reopening your drawing, it should remain checked ( I have yet to see it fail to display the tabs after this).
2. Do whatever you need to do, then activate "Clean Screen."
3. Notice that the file tab is missing (gasp!) and the "Options" dialog shows the "Display File Tabs" option as unchecked. By some "magic"/glitch/bug/whatever, it was forced off.
4. Close "Clean Screen."
5. Open the "Options" dialog again and notice that "Display File Tabs" option is unchecked. Check it and press APPLY and OK. No file tab will appear, but we have laid the foundation for the workaround. Note, if you were to close AutoCAD now and restart it, the file tab would reappear. Of course, that's not why I am writing this alternative procedure, as that has already been mentioned and if you have multiple drawings open, it quite frankly sucks to close them all and then reopen them. Instead, leave AutoCAD open and go to the next step.
6. The procedure at this step is entirely dependent on your workspace. If you utilize the "Ribbon" (the new style that utilizes panels of logical groupings of controls), then follow step 6A. If you are an fan of the "Classical" workspace, then follow step 6B. Since I recreated the "Classical" workspace from 2012 and imported it into the AutoCAD 2016 customization file, I do not use the ribbon, at all. Thus, I would follow step 6B.
6A. (Ribbon Users) Type RIBBONCLOSE in the command line. It will close your ribbon (resembling a "Clean Screen"). Then, type RIBBON in the command line and it will reopen your ribbon workspace. Notice that the file tab was restored! Strangely, now "Clean Screen" will also display the file tab and switching back and forth from "Clean Screen" will no longer hide the tabs nor force the option to be unchecked. If you were to close and restart AutoCAD, well, my friend, you are back at square 1.
6B. (Classical, Non-Ribbon Users) Type RIBBON in the command line. It will show the ribbon over your normal toolbars. Then, type RIBBONCLOSE in the command line and notice that the file tab was restored! Strangely, now "Clean Screen" will also display the file tab and switching back and forth from "Clean Screen" will no longer hide the tabs nor force the option to be unchecked. If you were to close and restart AutoCAD, well, my friend, you are back at square 1.
7. Enjoy the "workaround." I found that this persists so long as you keep AutoCAD open, so now subsequent evocations of "Clean Screen" won't screw up the file tabs until you close and restart AutoCAD (I am mentioning this again on purpose, as whatever bug is causing this has yet to be truly resolved).