What are the principal architectural similarities and differences betweel real-time and multimedia operating systems? What distinctive requirements does each have? What operating systems are typically used with each?
There is no short answer to your question and often the two terms are used together especially for multimedia OS.
One needs to understand the intention of the software here to answer your question here. There is no bullet item list possible. But to explain in short, I would like to say that "Real time OS may have multimedia support but multimedia centric OS need not specifically have real time features". Many may object this statement but consider from expectation or user experience perspective. Real time systems are designed for fastest possible response to an event or request. And that is the expectation. While multimedia os is designed to support multiple media. Now, unless one really thinks that multimedia means audio and video, multimedia need not be real time even if acceptable user experience is considered. Media can be looked in certain context. Support for different storage media like floppy, cd, dvd etc. can be considered as multimedia. From software perspective support for more than one media is multimedia e.g. text and different image formats such as .tiff, .jpg, .bmp etc.
Now, coming to generally considered meaning of multimedia - text, image, audio and video. Of this, for good user experience, audio and video need to be near real time (what Maria calls in her answer above as soft real time). Everything revolves here around user experience as far as multimedia capabilities are concerned.
Architecturally, multithreading, parallel processing, multiprogramming, dedicated memory for frame buffers, graphics co-processors at hardware level are desirable traits in multimedia OS. But for real time (consider for medical needs such as some critical health param monitoring equipment in ICU) clear data path on hardware for fastest processing is desired. There features like multithreading at software may or may not be needed but parallel processing, VLIW (very large instruction word), hardware multithreading etc. are desirable features at hardware level and support from real time OS to fully explore the hardware capabilities.
There is much more to talk about it but I guess, this will help you separate out two concepts with the thin boundary line especially from multimedia OS perspective. True real time systems are separate class altogether.