There were two CMI brands - this does seem to lead to a bit of confusion but they're completely unrelated.
In the UK, CMI was Cleartone Musical Instruments, which was indeed set up as a sub-brand of Marshall. The first CMI guitars weren't actually MIJ, they were rebranded UK-built Ned Callans (these also appeared under the Shaftesbury, Callan & Simms-Watts brands) Ned Callan was a pseudonym of luthier Peter Cook.
Subsequent Japanese CMIs seem to be from several different factories - including Moridaira, Kasuga & (possibly) Fujigen - which puts them in the mid-to-good quality bracket.
The other CMI was Chicago Musical Instruments - as far as I know they did not produce any CMI-branded Japanese copies, but there are some basic-looking original budget designs which appear under the brand. Chicago CMI is probably best known for being Gibson's parent company until the mid-70s, and this is what sometimes leads to some absurd claims being made about CMI guitars - particularly when they appear on Ebay!
Jon.