My client recently changed the identity of their company - from "Blackstone Edge" to "Blackstone Edge Films". The names are are similar enough that we kept their original domain (blackstoneedge.com), and redirected the new one (bsefilms.com) to the original. Make sense? When uploading Muse sites, I still use the ftp address of the original domain. In the "site url" field, I have entered "blackstoneedge.com, bsefilms.com" (including sub-directories, if we are creating a separate site.)
Everything seems to work fine on desktop browsers. We can visit the site (or sites located in sub-directories) at either blackstoneedge.com or bsefilms.com, and they work the same. However, on the iPad the sites behave differently. They are essentially broken if visited at bsefilms.com, but fine if visited at blackstoneedge.com.
blackstoneedge.com (original domain) - works fine
bsefilms.com (301 redirect to blackstoneedge.com) - broken on iOS, fine on desktop
I'm using some custom HTML on the main (root) site to make the slideshow at the top take up the entire browser window regardless of viewport size, so it might not be the best example. Still, it works very well on iPad at blackstoneedge.com, but not at bsefilms.com. I notice that when using bsefilms.com, we are presented with a different breakpoint than when using blackstoneedge.com.
There are other pages that have even worse issues when using the redirected domain, but I'd have to PM them as they are proposals my client does not wish to share publicly.
From what I gather, the third-party HTML code seems to be messing up your sites. Can you try removing that piece of HTML code for testing purposes, and see if there display is any better on an iOS device?