Qt: Show popup menu, close if I click anywhere outside of it - c++

Question

I want to show a menu in Qt with the following functionality: clicking anywhere outside of the menu will close it, but the click will be processed by the receiving widget. As far as I know, this rules out using setWindowFlag(Qt::Popup) because the click that closed the popup will be 'eaten' and not processed any further. I'd love to know if there is a way around this.
For the time being, I'm overriding focusOutEvent like so:
class Menu : public QWidget
{
Q_OBJECT
public:
Menu(QWidget* parent = nullptr)
: QWidget(parent){
setWindowFlags(Qt::FramelessWindowHint);
resize(200, 100);
}
public slots:
void showMenu(QPoint point)
{
move(point);
show();
setFocus(); // so that focusOutEvent will be trigerred
}
protected:
void focusOutEvent(QFocusEvent *event) override
{
hide();
}
};
And I'm using it like this:
class Window : public QMainWindow
{
Q_OBJECT
public:
Window(): m_menu()
{
auto* list = new QListWidget();
list->addItems(QStringList() << "item1" << "item2" << "item3");
connect(list, &QListWidget::itemClicked, [this](){
m_menu.showMenu(QCursor::pos());
});
// Layout stuff
auto central_widget = new QWidget();
auto layout = new QVBoxLayout(central_widget);
layout->addWidget(list);
layout->addItem(new QSpacerItem(20,200, QSizePolicy::Minimum, QSizePolicy::Expanding));
setCentralWidget(central_widget);
}
public:
~Window(){}
private:
Menu m_menu;
};
Single clicking on any of the items in the list will show the menu and clicking anywhere on the grey background will close the menu.
The issue I'm facing is that when the menu is shown, focus is lost from the main window (photo on the right, the titlebar buttons and selected item are greyed out). Focus is lost because of calling setFocus on the menu but if I don't use it then the QFocusOutEvents are not trigerred and the menu doesn't close.
The same happens if I use setAttribute(Qt::WA_ShowWithoutActivating) on the menu.
So here are my questions:
Is there a way to use Qt::Popup without the 'eaten' clicks issue?
If not, is there a way to show the menu without losing the focus from the main window but still trigger the focusOutEvents as necessary?
Using Qt 5.15.1 on MacOS 11.2.3. Thanks for reading through my long post!