living in the northwest, definitely not unthought of to fender up the bmx.
had no idea about water up the back until it happened to me while riding in rain. except on a bmx, it's your ass that gets the soaking, not a streak up your back. presumably because you do so much riding off the saddle on the bmx.
i'm now curious if the ass gets wiped riding fenderless on a road bike. haven't taken chances.
i have an sks quick install fender for the rear. it's made for mountain bikes and it's a bit big, kinda goofy but gets job done. there are probably less obvious ones you can get that are equally convenient.