If you have a runny nose, the standard procedure in Japan is to just keep sniffing or find a bathroom or secluded place to blow your nose in.
Blowing noses in public is pretty rude, even though handkerchiefs are pretty popular in Japan. However, people use them to wipe off sweat in the summer or dry their hands in public restrooms. Many public restrooms in Japan don't have hand-drying facilities surprisingly, despite having space-age toilets that reside in the same room.
Even though blowing your nose is rude, it is usually okay to do it if you do it quietly. The main rule is to avoid making excess noise that catches attention.
Coughing and sneezing are tolerated because they are sometimes inevitable, but you have to turn away and keep distance from people when you do so; this same rule applies to blowing your nose.
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Things Not To Do In Japan
Non-FictionThese are some cultural norms you should never do in Japan if you are planning to travel or move to Japan. I have never been to Japan myself; this is just information I found online from people living in Japan. I give credit to Abroad in Japan's vid...