I believe there is an issue with the widgets you are using, namely the AppSwitcher and the ProfileMenu.
They are both loading and calling the same service. Moreover, they appear to be trying to issue a cross-domain iframe call, something that the browser blocks for security reasons.
All in all I think there is some issue with the widgets, or in the way they are configured. Maybe they are not compatible with one another.
Someone seems to have had a similar issue - https://github.com/mendix/IFrame/issues/2
PS: An explanation why the app behaves differently locally from the cloud, is that the cloud runs under https, and is therefore allowed to invoke other https iframes.
Two things come to mind which might be very obvious, but did you set "Allow anonymous users" to yes in Security?
Further, are you able to create a branch of a version of a few days ago? Possible under "Team" and then "Manage branch lines"
Another reason why a login pops up is because you are trying to show something to the anonymous user it is not allowed to see. So double check you security settings to see if the anonymous user has access to all the data you want to show him.