To be clear, you can cross a river into an enemy-held region in PP[SUP]1)[/SUP], which will not allow your force to automatically change to OP and attack with the Amphibious Assault penalty. Once that force has arrived it might regain a small amount of cohesion--cohesion recover has enemy MC factored into it--, but it will not gain any MC nor entrench.
Once the force which has 'sneaked' into the region changes to DP--since its faction has <5% MC--it will automatically change to OP and attempt to attack any enemy forces in the region.
IIRC the Amphibious Assault penalty--ACG is correct; it only occurs if your faction has =<10% MC in the region--is only during the first round(s) of battle--during the actual landings ("We shall fight them on the beaches!...." -
Fat-Guy with a cigar) , so a 'defending' force, which is threatened to be overwhelmed, could be set to Offensive Posture with the Defend and Retreat RoE (Rules of Engagement) to only fight in the first round(s) of battle--where the Amphibious Assault rules will give it the greatest chance of doing damage--and then skedaddle.
[SUP]1)[/SUP] Of course, if your opponent has anticipated such a move and is in OP your crossing force will be in a world of hurt
