I have the scatterplots available now, though poor quality images.

The top plot has particle hidden variable angles going from 1 to 360 degrees along the horizontal axis.

The vertical axis shows the difference (or 'error') between the exact projection onto Alice's vector angle a , where a is zero degrees. The discontinuities arise when the exact projections are near zero and change sign as the particle angle increases. Eg from +0.1 to -0.1. But the quantised projection flips from +1 to -1 or vice versa causing a discontinuity on the graph as the errror flips from +1 to -1.

The middle plot is for Bob where angle b for Bob is 45 degrees. So the plot is similar to the Alice plot but moved 45 degrees to the right.

The bottom plot shows the scatter of Alice's error (vertical axis) against Bob's error (horizontal axis). A product moment correlation of -0.2 doesn't do it justice. The correlation IMO is locally for a smooth, though segmented, line/curve but globally the segments are unusual :)