Flat maps are simple and they work.
If you want the civ map to be a true globe and still view it as a flat screen divided into sectors (squares or hexagons or actually any polygon you like) you will have the same problem as the cartographers. Something has to break or be distorted.
From the practical point of view, turning civ maps into true 3D globes means a lot of work so I am not holding my breath.
However, this doesn't mean you couldn't have globe-like behaviour with flat maps. With appropriate edge-wrapping you can get more interesting maps, and wrapping isn't any big issue to implement with computers.
My vote would thus be: leave the maps flat but include different wrap-options for them settable in the editor. For example:
1) Pancake: No wrapping. Flat-Earth society is right and Ol' Cristobal would have dropped of the edge.
2) Vertical Cylinder: East-West wrapping. This is the current civ map.
3) Horizontal Cylinder: North-South wrapping. Like 2 but rotated 90 degrees.
4) Donut: East-West wrap plus North-South wrap. This isn't a globe although many games use it as such. The actual topology is more like a donut.
5) Flattened Balloon: This object has the same topology as a globe and would be a better approximation of Earth than a donut. Except it produces a weird-looking map.

Imagine it like this: Start with a square disk. In the midpoint of the topside draw a dot and label it North Pole. Same with the downside except that's the South Pole. With this image in mind the wrap algorithm should now be obvious: Start from the North Pole and travel to the edge of the square disk. This is the equator. Cross the edge to the downside and travel to the downside midpoint. You are now in the South Pole. Continue in the same direction (except it isn't the same direction anymore because you are now heading North

) until you again come to the edge. Cross it to the topside and travel to the midpoint (North Pole) and you have succesfully gone round the world. Nice, but as I said, the map will look weird...
6) Polar Displacement: As mentioned Earth is not a donut and you shouldn't get to the South Pole going North when in the North Pole. However, if you look at an actual globe Earth Map and what happens when you cross the Poles, you should easily figure out what ought to happen in a similar situation with a flat map: going north when you are in the top (northernmost) row of the map should move you one-half map width to East/West. Of course, as the civ map does not take into account that squares near the poles are smaller than those near the equator this again produces a weird result. But it could be fun or at least sometimes surprising.
