I don't think I'm going to be able to make a great bug report for this one, but I think last night I fought my own fleet.
I had just started the game, and I had two ships (took the starting option to have a second officer). I ran into a pirate fleet with the same type of ships, and I noticed specificially that the pirate fleet's second ship had the same name as my own second ship, The Third Moon of Mars. When I got into the fight, the pirate ships had no health bar; they showed as if their hull was already down to nothing.
Combat went on for a short time, and then suddenly I exploded. The fight was pretty quick, and I'm certain my own shields never went down. I think every time I hit the pirate ship, my own health went down. Maybe some kind of problem that crosslinked the datastructures so that my ships were in both fleets? I don't know; that doesn't really make much sense. Certainly the positions of the ships were independent. It was late, and I went to sleep instead of filing a bug.
What I do know for sure is that the pirate ship had no health, and couldn't be damaged.
Should there be anything in the starsector.log about fleet combats? I see lots of trade fleets, and system bounty probabilities, but nothing about the actual combat.