Keep palette visible when Mathematica application loses focus

I am on Windows and palettes go away just like you describe. It gets fixed by changing the default WindowFrame option away from "Palette"

EDIT

It seems that changing WindowFrame disables the window floating behaviour also when Mathematica has focus. So, this would be equivalent to setting WindowFloating to False. This last option would at least keep the "Palette" looks.

If you try creating a dialog instead of a palette, and setting its WindowFloating property to True, then this behaviour also starts happening. So this seems quite unavoidable and buggy

In the Option Inspector there is the option Notebook Options >> Window Properties >> WindowFloating. If I make a palette editable and set this option to False, the palette loses the behavior you want to suppress.