I'm a networking professional and have a really dumb question that at this point I'm too afraid to ask. Can a large company use the same public IPs in different locations?

It just occurred to me today I don't think there would be anything wrong with using duplicate IPs across the internet for mirrored sites. And I'm wondering if that's even standard practice now that I think of it. I see a problem if you don't have out of band management, but if you do...

Let's say you have a site in L.A., and a site in N.Y. They're redundant sites, and the servers are exactly the same. Can you put 1.1.1.1 on the server in LA, and 1.1.1.1 on the server in NY and just let BGP calculate the best route? And is this ever done in practice? Aside from management reasons, I don't see any reason that wouldn't work (unless maybe, just maybe, a user is smack in the middle and getting their packets "load balanced" between the two sites). Purely from a user's perspective, this would work, right?

Sorry for the dumb question. I've never once considered how duplicate public IPs would work out until today. And it's kind of blowing my mind.