This achievement is earned after the game over screen when Round 4 is cleared for the first time. The game need not be started from the beginning. The only conditions are that the round must be cleared and that the achievement is not awarded until after the game is over:
New Rally-X allows you to start the game from any round that you have reached. So, if you have reached Round 4 but not cleared it, you can start a new game from Round 4. When you clear it, you earn the
Clear Round 4 achievement but you do not unlock the achievement until after the Game Over screen.