Originally Posted by Don Mueller

Maybe there is a miscommunication.
When you say 'runs around F5' that seems to imply that he ran around to the outfield side and then went home. If R2 was judged not to exceed the 3 feet as he ran around then it is not an automatic out. The fact that he missed third while F5 was making his throw to F1 means nothing until an appeal is made.
If however, R2 veered to the inside when F5 attempted the tag and then continued directly toward home it seems fair to rule he went beyond his 3 feet and calling R2 out for leaving the baseline is a bit more obvious.