Why don't they just write a thin layer that interfaces Windows library calls to native Linux calls, possibly with a thin wrapper to shim it where nothing else quite fits? You could re-implement maybe some of the Windows libs as a "clean room project", given the functional description, that then hands off to an equivalent in Linux.

There's probably a cool and snappy acroynm you could use, like Virtual Interface (with nearly) No Overheard, or "Vino".