Just throwing this out, if turning of this houses' electrical service did not change anything, how about turning off the neighbors' service?
If all underground water lines are tied together, and are metal, and someone has a faulty electrical water heater, it could throw voltage onto the water line, traveling to the outside spigot at a different location.
Of course, all that has to be in place, but I have seen it.
I had this issue twice a number of years back. First time was with exterior (ground mounted) floodlights. The final straw was when we measured 14v from the pad mounted xfmr to earth. Utility opened the xfmr and found that there was no ground rod. Installed one and all was well. Second time I had this was in Forsyth GA. All utility was ok but we suspected one of the neighbors was backfeeding to the water pipes (whole neighborhood was metal piping). We actually pulled the meter on our customer's house and still had the problem so we knew it was from elsewhere, not from our customer. Utility could not find out which neighbor had the problem. We had the plumber take out a section of the metal pipe (between the water meter and the house) and replace it with plastic. That solved the issue for our homeowner. Never did hear if the utility found the problem house and never heard another word from the customer.