DesqView, at least in its earlier incarnations, ran fine (if slow) even on an 8088. Eventually they moved to /386 and /X. It's kind of a shame that the /X version didn't gain traction, but by then there were other options ranging from Linux to OS/2 to Windows 95.

There was another oddball multitasking OS out there from the late 1980s as well: VM/386. It was originally developed by Softguard (better known for floppy-disk copy protection schemes) and then sold to and launched by IGC. It used the 386's virtual 8086 mode to run virtual machines, a far predecessor to VT-x and AMD-V. I never saw it in action, though. It ran separate DOS instances in each VM, but the 386 only supported real-mode VMs.