What is causing these water "trails" on inside of exterior wall?
I wanted to comment on this but don't have enough reputation.

My childhood house had these marks in the basement on interior walls. They looked just like the marks in your picture. They only occurred on walls near my bathroom where people generally took long, hot showers. 

I'm not saying this is the cause, but for me they clearly had a correlation with the steam from the shower. The more long and hot showers were taken the more visible the marks seemed to be. 

Again, not really intending this as a good answer, but I cannot comment.