On some of my hardtail guitars, there's enough string length between the bridge saddles and the tailpiece that the string can vibrate on both sides of the bridge. On some notes, the string length between the bridge and tailpiece is close to being tuned to a harmonic of the note being played on the neck side of the bridge. Because the two notes are not exactly in tune with each other, there is some warbling produced as they 'beat' against each other. I discovered this while I was doing some recording a few years ago, and I decided I liked it for some tracks, but it was unwelcome on others.

Try stuffing a piece of rag under the strings on the tailpiece side of the bridge to see if the phantom notes go away. I found that weaving some heavy curtain cord through the strings will damp out these notes, and I can easily remove the cord when I want to allow them to ring.