Heroes of Might and Magic 3 or 4 (persistent world mode)
Solution 1:
What you are looking for is called "Play-by-mail" mode. Unfortunately, quick search didn't yield any results for HoMM. But if you check this list on Wikipedia Play By Email Video Games, there are some similar turn-based games with this feature, and the most notable are Civilization series and Age Of Wonders series.
And for HoMM you already know the solution:
sending (e.g. mailing or whatever) each other an updated savegame each time