This sequence is also shown in a gameplay video.
Move to the edge of the ledge and jump to grab the long, skinny vertical pole close to the side of the building. Here Alister begs Lara not to look down. (Bet you did anyway.)
Climb almost to the top of the pole. Then jump to grab the horizontal bar attached to the back of the next sign.
Lara's weight rotates the sign 90 degrees.
Still hanging from the same bar, turn around, move to the right side of the bar (toward the building), swing around and jump back to the vertical pole.
Climb up a bit and jump onto the ledge, which is now horizontal since you turned the sign. Cross the ledge and jump onto the next small ledge ahead, which is CHECKPOINT 14. (Is anybody else wondering what the people down in the street are thinking about all this?)
The final stage of the climb consists of a vertical pole followed by 4 horizontal bars. These are unstable, so timing will be important; however, the pole won't actually start to break until Lara gets to the top. So just jump to grab it and climb straight up without shifting Lara's angle on the pole. You can tap Interact to climb faster if you like, but this isn't necessary.
As you climb, the camera will start to shift to the right. Let it.
Then when Lara does reach the top, the horizontal bars will be directly behind her, and the camera will be pointing toward them. When Zip shouts a warning over the headset, immediately press Jump, without any direction keys or movement on the control stick, and Lara should jump back. . .
. . . and grab the first horizontal bar.
The camera should then align itself behind her. (If it doesn't, shift it with the mouse/right stick or just tap the HUD key.) Once Lara has grabbed each bar, she'll have time to swing around twice at most before the bar breaks. So jump to the next bar as soon as you can.
Quickly swing, jump and grab across all 4 horizontal bars before they snap. It may help to hold the Forward key (or press the left stick forward) throughout. Then Lara will begin to swing around each bar as soon as she grabs it. On the downswing, press Jump to leap and grab the next bar. If she flubs a grab, quickly press Interact to save. Then complete the next swing and immediately jump. When Lara reaches the fourth bar, swing and jump once more to the balcony ahead.
[Part 1 | Return to Japan Walkthrough]