When moving around the sites on Stack Exchange using Safari on my iPad (not the Stack Exchange App), I am fine when moving to any site other than SuperUser. When logged in, I can switch sites and stay logged in, but when I switch to Super User I need to log in again.
If I choose not to log in and switch to another site, I am logged in there.
Is there a problem with the login handling for Super User?
If it helps, I log in with a Stack Exchange account, not through Facebook or Google.
After getting fed up with the problem, I took the advice given in the answer below which was to use Chrome, and the problem still exists. The same goes for Firefox. Have I missed something? As you can see from the screenshots below, there are no cookie settings to be found.