Hey Dan,
Well.. I'm not really the one to ask on this one, because I hate pretty much everything about Apple's implementation of Desktops – and I only rarely use them in a very limited fashion.
That said – I've spent a lot of time investigating same for various users over the years.
System Events cannot see into desktops that are hidden – only the current one – so that option is out.
Something like this would normally work, but it seems to fail with the Keyboard Maestro Engine – at least with a test Text-Display window.
tell application "Keyboard Maestro Engine"
name of windows
end tell
Hopefully some code that @CJK wrote can come to the rescue:
ObjC.import('CoreGraphics');
let wInfo = $.CGWindowListCopyWindowInfo(16, 0);
wInfo;
I suggest you review this topic:
How to List All Windows of One App That Are Open in All Desktops - #5 by August
I will not bet the farm that you can actually activate the given HTML-Prompt window, but you should be able to detect it at the very least.
-Chris