... that God loves us and wants us to be happy. -Benjamin Franklin-
The English rules include this in a Questions and Answers section:
Move and Battle
Q: I move into a Province ‘At War’ occupied by Legions from
several opposing players. Whom do I battle?
A: Choose one opponent and battle. There can be only one
Battle per Province (or Sea) per turn.
I believe the "one Battle per Province" means once for the active player - in your example Rome would choose to fight the Greeks or Egyptians but not both.