Hmmm, years ago I had a friend bring me a car (a 1970s vintage Dodge) that had a similar problem. Turned out that whoever had worked on his car at some point in the past had forgotten to reattach the ground strap between the body of the car and the engine. I hooked it up and presto, problem solved.
Sold: 1998 DGC Sport 3.8 (Final odo: 178,000 miles)
Sold: 1998 Chrysler T&C LXi 3.8 (Final odo: 190,000 miles)
Sold: 2003 DGC ES 3.8 (Final odo: 172,000 miles)
1999 Chrysler T&C Limited 3.8 (Current odo: 190,000+ miles)
2001 Honda Accord EX-L V6 (4-Speed automatic)
2012 VW GTI (6-Speed manual)