Meeting Somewhere on the Grid

You work at corner A of a street grid and your friend works at corner B, which is 3 blocks east and 5 blocks north of A. Neither of you settles who walks to whom, so at the same moment you both set off toward the other's building along a shortest route, one block per minute, at the same speed.

At each intersection you flip a fair coin: heads you walk one block east, tails one block north. Your friend flips too: heads one block west, tails one block south. When only one direction still lies on a shortest route (you have already used all 3 of your easts, say) that step is forced and no coin is flipped.

What is the probability that the two of you stand at the same intersection at the same moment? Answer format: round to 3 decimal places.