a song of ice and fire - How did the Shy Maid pass The Bridge of Dream twice?

Minor 'A Dance with Dragons' spoilers ahead.

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While crossing the Rhoyne, the Shy Maid encounters some trouble in the ruins of Chroyane. Just before the attack of the living slaves (to Greyscale), the Stone Men, something rather odd happens.

'The Bridge of Dream,' said Tyrion. 'Inconceivable,' said Haldon Halfmaester. 'We’ve left the bridge behind. Rivers only run one way'

_{A Dance with Dragons, Tyrion V}

I haven't the foggiest what happened here. From my reading the Shy Maid ends up passing the Bridge of Dream twice, is it known how and why this happens?

Answer

This scene is a remnant of a larger chapter than GRRM excised from the book. It's meant to be spooky and confusing, and is almost certainly the result of magical interference.

During the trip down the river, Tyrion thinks about a mysterious creature called The Shrouded Lord. In the novel, we never see this person, and Tyrion starts thinking of him as a symbolic representation of his father.

From what I've heard of the original scene, Tyrion meets the Shrouded Lord; he is similar to other legendary beings we've met and/or heard of, in that he can make things happen that appear magical. In this case, he really did make the river behave unnaturally without anyone on the boat seeing it.

Martin claims he didn't like where this particular interaction took Tyrion's character, so he took most of the scene out:

When that happens, maybe my heirs will decide to publish a book of fragments and deleted chapters, and you'll all get to read about Tyrion's meeting with the Shrouded Lord. It's a swell, spooky, evocative chapter, but you won't read it in DANCE. It took me down a road I decided I did not want to travel, so I went back and ripped it out.

However, remnants of the unnatural nature of the river are left in, making the scene intentionally surreal. Besides the obvious fact that Tyrion passes the same point on the river twice (not just the Bridge, but the entire surrounding scenery), there's also this remark from Yandry:

"Mother Rhoyne runs how she will" murmured Yandry

_{(A Dance with Dragons, Chapter 18: Tyrion V)}

Rhoynish magic is all about water magic; this is intended to evoke that kind of idea that the river itself is magical, though the reasons why it happens are left to the imagination.

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There's a lot of speculation about what was in this original chapter; while we'll never know, there are some other narrative bits that seem to go nowhere that may have set up this encounter. First, we have this bit about The Shrouded Lord:

They say that the Shrouded Lord will grant a boon to any man who can make him laugh.

_{(A Dance with Dragons, Chapter 14: Tyrion IV)}

We also learn that The Shrouded Lord might be

The still living prince Garin,

and might be the cause of greyscale, as described in The World of Ice And Fire:

And so, that very night, the Rhoyne flooded out of season and with greater force than was known in living memory. A thick fog full of evil humors fell, and the Valyrian conquerors began to die of greyscale.

_{(The World of Ice and Fire, "Ten Thousand Ships")}

One popular theory says that, in the original scene, the Rhoynish magic of the river was reacting to the presence of

the Valyrians -- specifically, Aegon

on the boat, making them sail under the bridge a second time so the stone men could attack. After being pulled underwater, Tyrion meets the Shrouded Lord and makes him laugh. In exchange, he's granted a boon" of not catching greyscale, but only if Tyrion

talks Aegon out of seeking out Dany for himself.