The rivalry is still VERY real when it comes to the Sox and the Yankees ... just ask Rob Kardashian who tells TMZ Sports he almost got his ass kicked in Beantown for rockin' his Yanks hat.

Rob says he's still a Dodgers fan at heart -- after all, he was born and raised in L.A. -- but says his lady Blac Chyna got sick of the L.A. lid and pressured him to switch it up. 

So, Rob chose the pinstripes ... but forgot how much the Yanks are hated in Red Sox country -- and says his fashion choice almost got him beat the hell up.  

Hey, it's rough out there.