This is interesting. I have had this happen many times in the past because I played songs that people knew. My instinct was to slow down and follow them, that is the accompanist in me I guess.
A weird one that happened was I was playing before a wedding. I have an arrangement that I did of Chi il bel sogno di Doretta from Pucinni’s La Rondine. Not many people know this beautiful aria, but from behind me, as I was playing, this woman began to sing, what a lovely voice she had! I continued to play and she sang never missing a beat!