Stream cipher with an additional integrity stream?
Break your stream into chunks.  Use an AEAD for each chunk using the chunk sequence number as a nonce.  Anything else is going to have at best the same effect, but will be convoluted, full of holes for the first umpteen drafts, difficult to implement, and nonstandard, and will have the effect of making cryptographers look at you like you have three heads.

Or, use a system that was already designed to do this (in particular, STREAM).