Clusters H7 and H8 at various depths

The blue tiles in the first two images are the only flipped-over hat tiles in the two clusters.

A long slightly decreasing sequence of aligned H7 and H8 clusters running from left to right is highlighted in the images.

Substitute a sequence of 887 clusters along that chain with a 0 and the remaining sequences of 87 clusters with a 1 (7 stands for an H7, 8 for an H8) and you obtain a chunk from the Fibonacci word.