I feel like "The Cohost Global Feed" is too long and formal of a tag and could be workshopped a bit
First off, the use of capital letters is too stuffy IMO, but we can't really change that at this point. However, I'll be using all lowercase for the remaining suggestions.
One of the things I'd focus on is getting rid of some of the redundancy from the tag name. Do we really need the "the"? No. We can shorten it to "cohost global feed".
Also, do we really need to specify that this feed is for "Cohost"? Not really. It should be clear to everyone using this website that this feed is located on Cohost, so we can shorten it further to "global feed".
Now, we could in theory drop the "feed" since all tags on Cohost are feeds so that is a bit redundant. The problem with that is that a tag like "global" could be used to describe other things (like international news, for example), so it's a little too short. Also, a tag called "global" wouldn't be self-explanatory. Ideally, you'd want someone who sees this tag show up on their timeline to intuit what its purpose is for, and "global feed" works better in this regard than "global".
If you really wanted to workshop this further, you could look for alternative names for "global" and "feed". For example, you could call it the "global timeline", although I personally think "global feed" is better because there's already prior art for this name (on mastodon) and it's shorter. It also benefits from being similar to the "The Cohost Global Feed" tag.
I'm obviously not the first person to think of this since the "global feed" tag already exists and has quite a few posts. It seems like a lot of people posting to "The Cohost Global Feed" have already been cross-posting to "global feed". However, if there is an origin story for the "global feed" tag then I can't find it (since there's not an easy way to browse the earliest posts for a given tag).
Anyway, this is just a long-winded way of saying that I prefer the "global feed" tag and I think the "The Cohost Global Feed" tag is noise and tag debt.