When it is not your turn, you are able to do things such as look at your tech trees, design ships, or give build orders to your colonies. When your turn comes up, however, if you do not hear the ping indicating your turn has arrived, there is no way to know it is your turn without returning to the galaxy map screen.
There should be a "current player turn" such as that on the galaxy screen but present on every screen during multiplayer gameplay.
Might be cool to give human players a "+5% economy" boost when their actual turns takes less than some specified time, such as 1 minute. (Specifics obviously to be decided.) This would encourage fast turns and planning during other player turns.