I was trying to search for it here on Google and didn't find it, but found this and was shocked.
Complete Neo (Keanu Reeves) Costume
This is the entire thing, which I have NEVER seen before. The thing that interests me the most after the actual coat is the shoes he wears, which are more like boots without laces. In that link, the shoes have laces, but I definitely don't think it's correct.
Here are two screencaps. In the first one, you can clearly see there are no laces, and in the second one (in the lower right corner) you can see that they go above his ankles.