Still planning to perform this experiment today, but yesterday I was at the Griffith Observatory in the afternoon. And it was a little before sunset when we were leaving when I decided to just take out my shoelace and give the string test a preliminary try. The sun was getting low to the Hollywood Hills, but it was still high enough and bright enough to cause sighting problems.
So I used a sign to block the sun, had my son hold one end taut at where the sun was occluded behind the sign, and I stretched it straight to line up with the moon (now in its 1st quarter).
This wasn't ideal since I found it very hard to capture the scene. Even in the fullest view, the sun and moon are at too wide an angle to get in a single shot without a distorting wide angle lens. Also, when zoomed out to show the length of string, it makes the moon too small to see its terminator. Zooming in, and you lose simultaneous focus on the string and moon. I tried different aperture settings and focal lengths, but I was limited in time so this is the best I could do. Hoping for better performance this evening under a more controlled setting, but this might serve as a precursor as to what to expect.
Here's a crude "panorama" of the moon-to-sun, which formed an angle greater than 90°
I added a little inset zoom of the moon to help make the terminator line more visible. But lacking index/reference points between images, I can't be sure I've aligned the separate images correctly. In fact, I'm almost sure I haven't since the moon should be higher relative to the flat lateral span traced from the sun. Still, even when skewed lower I think the illusion is still there since the shadow line on the moon still looks like it is perpendicular to a line that points higher than the sun. Right? Looks like it's pointing toward the upper right hand corner of the "panoramic" composite. And that's how it looked in real life too. I needed to explain to several curious people who asked what I was doing and all agreed that it did look like the moon's terminator wasn't aligned with the sun. So the illusion is real.
But looking across the stretched string, which I avow was straight and taut:
That's my hand holding the string aligned with the moon straight out and up in front of me. My son's hand is holding the other end which was about level with my head and off my right shoulder.
It lines up: