In fact, quite a lot did.
Ignoring the 'home computer' restriction there is of course MP/M - the multi-user and multi-program environment for CP/M. MP/M was published in 1979 by Digital Research for 8080/85/Z80 machines. Terminals, users and programs were handled separately, thus one user could have several programs run in parallel and switch between them (called "detach" and "attach") on a single terminal, or change terminal and attach from there. Also, several users could (in sequence) share one terminal. Programs could run attached (in foreground on a terminal) or detached (in background). In addition, there was a scheduler process starting (and stopping) programs at specific times and conditions (like cron). Last but not least there was a separate spool process, so programs were not blocked while printing.
MP/M also included functions for inter-process communication (queues) and process synchronization. Soon a network level (CP/NET and CP/NOS) was added to connect multiple MP/M machines to a kind of cluster (very rough term, but it was more than a simple client/server structure and I don't know how to explain its workings in less than a few pages :))
All features could be controlled using the MP/M extension of the CP/M API.
In theory, MP/M could have been used on every computer capable of running CP/M, especially where CP/M 3.0 memory management was available (which in itself was a backport from MP/M II to CP/M), but other than some Tandy Model II and 4, I don't remember any home computer with MP/M support.
Another very common multiuser/multiprocess system was OS/9 created for Motorola's 6809 CPU. Since the 6809 offered great support for position independent code and data, as well as OS support and module linking in Hardware. Thus it was easy to load not only several programs at once, but also to support re-entrant code, thus resulting in a great reusability for libraries (shared code).
Another multiprocess OS for the 6809 was UniFlex. Flex was originally written as a single-user single-program OS for the SWTPC 6800 machine. Later iterations included a port for the 6809 and integration of Unixoide functionality, then called UniFlex.
For home computers there have been dozens of variations of multitasking/multiprocessing environments. From 1984s M.O.S for Schneider/Amstrad computers, distributed by StarDivision to 1986s GEOS for the C64, a seemingly endless plethora of OS and OS-like environments have been created. I might need a book to list and qualify them all.
The most remarkable piece might have been the Sinclair QL from 1984. With a 68008 CPU, it might be seen somewhere at the edge, but I'd still consider it an 8-bit machine. The QL did include a pre-emptive multi tasking OS in ROM called QDOS. The built-in SuperBASIC did offer functionality to create and control processes. It is said that Linus Torvalds did take many inspirations for Linux from QDOS, as he owned a QL before switching to a PC.
(Oh, just to brag about: ca 1979/80 I wrote a small multi-process kernel for the Apple II, able to run up to 8 tasks, but I guess that's way below the threshold the OP asked for :))