Like other people have been saying, it was possible to stun phantoms in PH without the Phantom Sword. I'm not sure if you could do this with a sword slash, but I'm sure you could by shooting them in the backside with an arrow. If arrows work for that purpose, why not a slightly powered up sword?
As for the "but Bellum's not in it, how could there be phantoms?" question, phantoms might just be semi-sentient guardians of holy places that were created perhaps by Hylians, or perhaps by the "good spirits" mentioned in the ST prologue. Whichever, they would have had to be in both the PH world and the WW/ST one.
I haven't gotten a good look at it to know one way or the other, but what's to say the sword Link has in the game isn't the Phantom Sword? And where does it state that the Tears of Light go into the sword? I remember it stating that you need the Tears of Light for Zelda to possess a Phantom, not to make your sword capable of defeating them... :?
Eh, I doubt the sword that Link has in ST is the Phantom Sword. If it was, then why couldn't you kill Phantoms right off the bat? Besides, did Link even have the Phantom Sword at the end of PH, after he left the world of the Ocean King? I don't remember.
As for the Tears of Light, I don't recall anything saying that they would power up Link's sword, but what else would they do? I suppose they be what gives Zelda the ability to possess the phantoms, but I'm leaning towards the sword.