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So, Ultramar does raise Imperial Guard regiments, although I haven't found any specific mentions of which planets raise them.

I would guess Macragge, Calth, Talassar, maybe Masali, Tarentus and Quintarn, and possibly (although unlikely) Konor or Prandium (prior to destruction).

So, would they be designated by the planet of their origin (the Masaline 275th, the Talassarian 24th) or would they just be designated as part of the Ultramar system? (Obviously they wouldn't be called the Ultramarines, but maybe the Ultramaran 67th, or the Ultramarite 142nd?)

How would these groups be named, do you think? I haven't been able to track down any specific lore on them.

I found a little bit on some Calth regiments, although they have some weird names, so my question still stands.

In the Know No Fear book it has some regiments from Ultramar, the only one I can remember off the top of my head is the Numinus 61st.
Numinus city is (was?) the capital of Calth, so maybe planet and/ or city names are the way to go. I'm not sure if much will have changed in Ultramar over 10,000 years.