Each military would have it's own code name for the offensive. This is also a good way of keeping the other guy on a backfoot.
If 4 or 5 militaries are all preparing for operation "morning glory" then the other side's intell uits can make a pretty good guess that something big and unified is going on. If 4 seperate peration names are being used then the other guy has to consider if it's one big op or several smaller, local ones.
The way I'd write it is to give the offensive a codename for each military taking part. Everyone involved knows what that codename refers to.
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