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Though right now I'm playing on a map that doesn't have a sailing route along the Northern Border and it "feels" pretty realistic but then I'm still in early days.
Civ4 had the visual thing where scrolling backward far enough showed you a globe, but those worlds weren't spherical when actually playing.
It simply is impossible with the current tiles. Perhaps if they went to triangular tiles it could be done? I'm not sure, that's not my strongest part of math.
The second option is to make the player interface into a globe instead of a square map. Back in the days when Civ was squares on a map, that was impossible. Now with hexes, it IS possible. But then you have to ask, what are the gameplay improvements compared to the drawbacks. One major drawback would be the entire user interface, having to scroll a ball around instead of a flat surface. More realistic - yes, more cumbersome and annoying - yes!
So that is your spread of considerations. Given that Civ is basically a beer and pretzels boardgame on steriods, flipping to a globe would be a break of genre. Maybe it is possible. Maybe it would produce good results. But the chances are low.
Rimworld dev did it and said he ended up needing to use a handful of pentagons towards the poles. Forget the exact number used.
Not sure how well that would work here, as in that portion of the Rimworld game there isn't much variation in the art. That would likely cause a MAJOR problem for Civ VI, as they have a lot of 3d rendered moving art for each tile. Not to mention possible pathfinding issues!
1) they faked it
2) it didn't matter
1) The vast, vast majority of the time you are playing the game, you are looking at a square map where you have a base, which is in reality a hex (so faked it it, a rectangle is NOT a hex).
2) As said above, the time you spend in the UI on the zoomed in hex, which is a square is probably 99,9%, which means doing a globe is of no consequence whatsoever. Try it when it is your MAIN interface!
However, there is a projection (don't recall what its official name is) that looks like an orange peel with splits (open spaces) where the the map is "cut" to flatten it out. With the way units seem to "teleport" when going through tunnels, it may be possible to implement it.
You are never going to sell that to the vast public. Some people might get it, and a smaller subsection might approve, but it will literally never sell.
The ONLY option is taking advantage of the fact that hexes can make an approximation of a perfected sphere. Which means UI problems, and AI problems galore.
For example, a soccer ball is made up of 20 hexagons and 12 pentagons.
https://en.wikipedia.org/wiki/Goldberg_polyhedron
The general category is described in that link. It should include all the basic Euler's polyhedron and the more bizarre stuff too. I'm a little rusty on some of this stuff though.
And apparently people at Firaxis made the priority list so that those items didn't make the cut.
https://www.ngdc.noaa.gov/mgg/image/images/etopo2icosahedron.pdf
This kind of grid is already used for some RPG maps :
http://inkwellideas.com/wp-content/uploads/2012/03/ico-map.png
Search for "icosahedron world map" to find more examples.