I worked at a place that developed a "Linux on Windows" thingy back in the Windows XP days. It was essentially like WINE. A user-mode Windows program would load the Linux binary into the Windows program's address space and execute it, trapping any attempts by the Linux code to issue system calls, and the Windows program would then service those system calls.
For non-GUI stuff this worked remarkably well. I was able to grab the binary for rpm off of my Red Hat system, and the then current Red Hat distribution disc, and install successfully almost all of the RPMs from the disc and have almost all of the non-GUI ones work.
I had expected big problems from the case-insensivite vs. case-sensitive filesystem issue, but in practice there were only a handful of things that ran into this. Mostly Perl stuff that used both "makefile" and "Makefile".
GUI stuff was another matter. We could run XFree86 under Cygwin, and then the Linux apps under our WINE-like program would work. However management was not keen on the idea of including Cygwin and XFree86 if we turned this thing into a project. Also, we wanted an X server that would fit in better with a mix of X and native Windows apps running at the same time.
I spent a while trying to write a Windows X server straight from the official specifications. I got as far as being able to get xcalc to display a window and all the controls to show up right, but weird things happened with events. Everything looked fine when I packet sniffed the communication. I still had not figured this out by the time management decided that this whole thing did not have enough of a commercial market to continue the project.