The DMG installer can be downloaded from here: https://github.com/aecreations/PyQtPopupDemo/releases/tag/0.0.2. It has been signed using my Apple Developer certificate and is notarized by Apple.

As for the use of QMainWindow - this is what I'm currently using in my actual app, which was changed from a QWidget after reading somewhere about tap gestures requiring a QMainWindow, though that change hasn't made a difference (and so I might as well change it back to QWidget).

Again, this is a sporadic issue, so you may have to try it a few times to cause the issue to occur. Switching between windows from other open applications and then clicking the menu bar icon for my app may cause it to happen.

demo.py:

import sys
from PySide6.QtWidgets import QApplication
from SamplePopupWnd import SamplePopupWnd

mainwnd = None

def main():
    global mainwnd
    app = QApplication(sys.argv)
    app.setQuitOnLastWindowClosed(False)
    mainwnd = SamplePopupWnd()
    sys.exit(app.exec())

main()

SamplePopupWnd.py:

import sys
from PySide6.QtCore import Qt, QPoint, QSize
from PySide6.QtGui import QIcon, QAction, QGuiApplication
from PySide6.QtWidgets import (QWidget, QMenu, QSystemTrayIcon, QHBoxLayout, QPushButton,
    QMessageBox, QMainWindow)

class SamplePopupWnd(QMainWindow):
    def __init__(self):
        super().__init__()
        self.popupwidth = 300
        self.popupheight = 400

        # Menu bar extra icon.
        icon = self.getSystemTrayIcon()
        self.systray = QSystemTrayIcon()
        self.systray.setIcon(icon)
        self.systray.setVisible(True)
        self.systray.activated.connect(self.togglePopup)

        self.button = QPushButton("button", self)
        self.button.clicked.connect(self.showButtonMessage)

        # Actions for a menu that is opened by a menu button
        self.settings = QAction("settings", self)
        self.settings.triggered.connect(self.showMenuMessage)
        self.about_act = QAction("about", self)
        self.about_act.triggered.connect(self.aboutActionSelected)
        self.quit_act = QAction("quit", self)
        self.quit_act.triggered.connect(self.quitActionSelected)

        self.menu = QMenu(self)
        self.menu.addAction(self.settings)
        self.menu.addAction(self.about_act)
        self.menu.addSeparator()
        self.menu.addAction(self.quit_act)
        self.menubutton = QPushButton("≡ menu", self)
        self.menubutton.setMenu(self.menu)

        hbox = QHBoxLayout(self)
        hbox.addWidget(self.button)
        hbox.addStretch(2)
        hbox.addWidget(self.menubutton)

        # Add to a QWidget that is styled with rounded corners.
        self.popup = QWidget(self)
        self.popup.setObjectName("popup")
        self.popup.resize(QSize(self.popupwidth, self.popupheight))
        self.popup.setLayout(hbox)

        self.setAttribute(Qt.WidgetAttribute.WA_TranslucentBackground, True)
        self.setWindowFlags(Qt.WindowType.Popup)
        self.setGeometry(200, 100, self.popupwidth, self.popupheight)
        self.setStyleSheet("""
            #popup {
                border-radius: 16px;
                background-color: #fefefe;
            }
        """)

    def showButtonMessage(self):
        QMessageBox.information(self, "demo", "the Button was clicked!", QMessageBox.StandardButton.Ok)

    def showMenuMessage(self):
        QMessageBox.information(self, "demo", "the Menu Action was clicked!", QMessageBox.StandardButton.Ok)

    def aboutActionSelected(self):
        QMessageBox.about(self, "about", "qt for python popup demo 1.0")

    def quitActionSelected(self):
        sys.exit()

    def getSystemTrayIcon(self):
        icon = QIcon("systray.png")
        return icon

    def togglePopup(self, reason):
        # Handle click on system tray icon.
        if reason == QSystemTrayIcon.ActivationReason.Trigger:
            if self.isVisible():
                self.close()
            else:
                self.openPopup()

    def openPopup(self):
        systray_geom = self.systray.geometry()
        x = systray_geom.x()
        y = 0 if systray_geom.y() < 0 else systray_geom.y()
        currscreen = QGuiApplication.screenAt(QPoint(x, y))
        screen_avail_geom = currscreen.availableGeometry()
        top = screen_avail_geom.y()
        left1 = systray_geom.x()
        left2 = screen_avail_geom.width() - self.popupwidth

        # Align popup left coords with the system tray icon.
        left = left1 if left1 < left2 else left2
        self.setGeometry(left, top, self.popupwidth, self.popupheight)
        self.show()
        self.activateWindow()

systray.png:

Do you have the same issue if using PySide6 ?
Can you share a minimal script that shows this issue ?