We finally got my visa! The process actually went quite smoothly. We booked a trip to Osaka for 8 days. We’d been told beforehand that the visa would take at least 4 days to complete. It ended up taking 6 full days, possibly a little longer than usual due to the fact that there was a holiday last week in Japan (Keirō no Hi).
The consulate in Osaka does not make appointments and was only open from 9:00 to 11:30 each morning. We were a little panicked and made sure to get to the consulate right at 9:00am the morning after we arrived in Osaka, but actually it was not very busy and we were able to get everything submitted and checked over quickly by 9:45am. They kindly do a sanity check (looking over your application for about 10 minutes or so) when you hand in your application, which was good, as I’d forgotten to write my address in a couple spots.
Thankfully no delays or anything. I will keep the visa application page on the wiki up to date, but actually it was a relatively smooth process. Next step now that we’re back in Korea: registering as an alien!