Workaround:
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Install Aurora Store from F-Droid (it’s what App Lounge’s Play Store part is modeled after).
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In Aurora Store’s Settings - Updates, enable Filter apps from other sources (“Don’t check for updates for apps installed from sources outside Aurora Store”), so that Aurora Store only shows updates for Apps it installed itself to avoid some common pitfalls (e.g. Aurora Store mistaking microG for Google Play Services and trying to update them, which doesn’t work but might be annoying).
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Force-stop App Lounge in /e/OS Settings - Apps - App Lounge, so that it doesn’t interfere once it possibly works again. This will probably not survive reboots.
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Once you want go back to using App Lounge, do it the other way around and force-stop Aurora Store, so that it doesn’t interfere.
There can be times when Aurora Store can’t access the Play Store in Anonymous mode either.
You can try to log out of Aurora Store and simply try again … cogwheel top right - Manage your account - Log out. If you are connected to a Wi-Fi network, you can also try whether leaving it and trying again via mobile network will help.
If the Filter apps setting is too restrictive for your needs and you want Aurora Store to update other Apps …