Why doesn't Zigler wear the IWGP global title when he wrestles is USA

Inter-company belt situations can be tricky.

A) Not everyone has the belt at all times. Sometimes the belt stays with the promotion (like WWE, outside of media appearances and special situations, will have the belts plus duplicates carried around the same as the ring and backstage stuff in a big crate).

B) Some companies may not want their belt shown around on other companies shows due to internal/backstage politics.

B) If you have the belt, especially the belt (if it's not WWE/AEW/TNA where they'd have a couple spares in the back) and you're going around on nation wide/international flights, driving around from city to city, there's a chance you can lose that belt. The belt that can cost thousands to replace. We've all seen cars broken into and wrestlers having shit stolen, luggage going missing, people forgetting the belts at random places, etc. If Black Christian brings the GCW Championship with him when he shows up on ROH/AEW, it looks fantastic. If he's an idiot and checks it in his luggage and that goes missing, he now owes GCW the cost of a new title or his arse isn't getting booked by them anytime soon. Some wrestlers are careful and aren't idiots, some aren't and can be.

Edit: I'm leaving it.

I also wasn't aware he had that belt