Interesting, I’m not sure why I can’t replicate this. I suspect it might be to do something with the timing.
Either way, I have a suspicion of what might be causing it. I see you have a child theme active, and I guess the other site may have the non-child theme active. So if I’m correct, then for some reason the switch of the blogs during the search is not reverting back correctly.
I have moved a piece of code via FTP so the switch happens sooner, but I can’t test it as it’s working on my end. Can you please see if you get anything different now?