A semi-date is an encounter between members of the opposite sex or of the same sex, whereby at least one of the aforementioned members is attracted to the other and hopes that the friendship becomes something more. It is referred to as a "semi-date" rather than a "date" because is has not been arranged according to the terms of a traditional "date", but rather happens out of the blue or is an instance of "hanging out as friends", during which the two of you spend time alone together.
I happened to go and see a play and bumped into this guy I knew there, whom I'm attracted to. Afterwards I suggested we go for a drink in a nearby bar, and as he didn't have enough cash on him, I paid for his drink. It wasn't a pre-arranged date, but we were alone, and I do have romantic feelings for him. You could say I paid his way in our "semi-date".