Is another strategic round played before battles are fought ?

When all players finished their rounds, the strategic round ends also.

Usually the next strategic round starts immediately, except:

If all but two players are resting or this was the third strategic round played since the first player became resting, than the battles are resolved before the next strategic round starts.

Designer notes.
All players meet at the same time. To prevent any of the resting players to get bored the number of turns between becoming resting and fighting battles is quite small. The maximum of three rounds that a player can not recruit  (compared to the others) lessens the disadvantage of becoming resting early in a strategic round.