Some kind of 'while you are waiting' you do some effort/activity which optimises your controlled game assets(objects you control), which is independant of actions taken by other players upon those same assets.
If you put extra effort in while waiting (keeping you busy and interested) then the delay isnt as bothersome (and the longer your opponent takes is MORE the benefit to you)
Possibly some short minigame-like operations (which can be continued if interrupted) withing the game objects.
Probably subject to 'decreasing returns' so that the delay doesnt bend the advanatages too much for any player
Example - I have a widget Factory on the map and I can 'help' it to be built faster by doing X manipulations and once in operation I can 'help' it product a bit faster (my decision which is priority to apply MY attention too) for something I consider most critical OR give broader attention to multiple assets (objects) which will (because of decreasing returns) reap the most benefit for the effort I expend (in that delay waiting time).
Of course it requires an additional; internal mechanism for the 'objects' to be (graphically) presented and interfaced with (managed0
Myself I always did wonder how well the 'online' versions of some games (like Civilization) actually worked when it took so long for 'your turn' to come back around again. I never did play them to find how they solved the 'sit and twiddle your thumbs' part of thatmechanism or what measures they took to prevent players from monopolizing the time (taking a long time to do THEIR turns) -- for games that arent long delays like the old postal games (which had a fixed weekly schedule)