A window with no taskbar icon, no appearance in Alt-Tab and *without* using the ToolWindow extended style

Question

I have a problem that appears to be new to Windows 10.

I want to create a form that is visible to the user, but with no task bar icon and that does not appear in Alt+Tab.

This is perfectly doable if one is happy to sacrifice the normal styling of a window by following the accepted solutions here for either WPF or Windows Forms.

The general advice for both WPF and Windows Forms is:

• Set ShowInTaskbar to false
• Enable the ToolWindow styling (either through setting the border style in WinForms or the WindowStyle in WPF)

However, this has a new, practical problem in Windows 10 when using Virtual Desktops: the moment you do the above, the WPF or WinForms window will appear in every virtual desktop. See my example application with a red background:

This affects both the Task View switching screen and the actual desktop itself. No matter where you go, the form is there!

Is there any way to show a form - or even just a bitmap - on Windows without anything appearing in the taskbar, without anything appearing in Alt+Tab and without duplicating the window on every virtual desktop?

I have spent two days researching every possible option, trying every example online, reading MSDN documentation on window styles etc. but all resort to the same method, either through P/Invoke calls or directly, but either way the result is the same.