Winter runny nose remedy?

A facemask may work once you dont overheat.

Bluntly, I hold one nostril closed, turn my head to the side, and blow the snot onto the trail behind me. Then I repeat on the other side. I find that much more viable than using a hankerchief. And I can do it while leaving my gloves on.

A face mask over the nose helps a little bit, but mostly your nasal passages are getting cold from breathing the cold air.

There are a few heat-exchanger contraptions I've seen at times. Some are marketed for reduced heat and moisture loss in extreme environments. Another went from a snorkel-like mouthpiece to a diffuse worn on one's chest under one's jacket. The idea was that one suffocates in a snow avalanche due to their mouth-nose getting plugged with snow but that over a larger area, snow could be porous enough to support breathing.

Look to what native peoples did/do – they have tunnel hoods on their parkas. The air inside with tunnel is really quite a bit warmer, glasses don't fog up as much, your nose and cheeks are warmer. It does limit you side vision a lot, but it works. I wear anything down to 0F. Between -20F and -40F, I use a tunnel hood. Any colder, and I stay inside as much as a possibly can.