I had two large orange drops of some unknown but very difficult to remove stuff on the top of my 2005 white car, along with splatter (indicating that this had dripped onto my car from above while it was parked), and a long streak of runoff down the back of the vehicle. My best guess was that this was some kind of terrible calcified/rust/paint mixture dripping through cracks in the ceiling of my parking garage.
I'm in the US and have not heard of the solution mentioned in the preferred answer. After reading this question/answer thread this past weekend, I tried a few things:
1. take car through a Sonic car wash. Did not remove stains.
2. manually scrub with "generic" car wash/wax solution and warm water, using sponge (nope), microfiber cloth (nope), and bristle brush (nope).
3. manually scrub with homemade solution of 2 tbsp baking soda and 2(ish) cups of warm water using microfiber cloth. With a TON of elbow grease, a tiny little bit came off, but not enough that this was a practical solution. This also left an annoying whitish film all over that I had to wash off with #2.
4. Add my father's urging (I didn't think it was going to work), manually scrub with Turtle Wax Scratch and Swirl Remover using a warm, damp microfiber cloth. He's had this stuff in his garage for god knows how many years and the result was magical. With a minimum of elbow grease, the stains came right out and the only thing left behind was a really nice wax job.
It also removed really, really old (like from years ago) paint transfer from other people opening their car (red, black, blue) doors into the side of mine. I thought those stains would never come out. In writing up the answer to this question I actually forgot the name of the product and in searching for it found a website that recommends the product for this very purpose: turtlewax.com
Full disclosure: I have absolutely no association with this company, I'm just a really pleased son of a customer. So if there is a rule against posting their URL, feel free to remove it. I found it to be such a surprisingly good solution that I figured it should be posted here.