On Windows.

Put a QWebEngineView somewhere inside QMainWindow. It even can be hidden, it just needs to be instantiated somewhere.

Put QMainWindow on the primary screen (where Windows Taskbar is located) and make it full-screen.

All menus, pop-ups and drop-down lists become invisible yet fully functional. They receive events and send signals when actions are triggered. They're just invisible.

It doesn't matter how to make QMainWindow 'full screen'. It can be a call to setWindowState(windowState() ^ Qt::WindowFullScreen), or it can be a call to setGeometry() provided with primary screen's full geometry. Both end up in equally corrupted behavior of menus, pop-ups and drop-down lists.

See attached project.