internship decision - Strong Fortunre 500

That's tough, exploding offer. Don't renege, make a decision and stick by it. By your asking of the question, I assume you want to do MM capital markets over industry corp fin and that you are a junior. If you're a sophomore, I'd be risk averse and take the F500 job and apply to banking firms next year. If you're a junior at a non-target, your chances of getting a banking capital markets job are probably not decisively high enough to turn down a lucrative opportunity with a good F500 that you already have in hand. Especially if you have fewer than 5 interviews lined up, since offer rates even from first-round interviews are slim. So if you're not close enough with this MD (i.e. he/she is not your parent or family friend), I think you should call the bank, request an accelerated interview process given your other offer, and if they decline, go with the F500. Obviously, talk to others before making this big decision. There could be some hope that with a brand-name finance internship for FT recruiting, you could still land a capital markets job, if that's indeed what you want.

Another thing to consider is whether you like the industry of your F500 offer and how well their rotation program feeds into business school/exit opps, assuming you will ultimately get a FT offer from the F500. Coming from a non-target, a brand-name F500 might be just what you need down the road. Strong b-school admissions could get you a quality BB or MM capital markets job later. Non-target + MM capital markets can be tough waters to tread as an analyst, since there is a bias towards analysts from target schools for buyside recruiting, although you might still land something

And lastly, "MM/maybe BB in some countries?" You got me playing the guessing game, haha, can't imagine what firm meets that criteria except maybe Asian firms in the U.S. (Nomura, HSBC). Good luck