This has been covered many times, and misunderstood almost as much. An asymmetric primary drive usually causes an asymmetric secondary voltage. There is indeed no DC in the secondary, if the load is resistive. However, a fluorescent tube isn't resistive. The tube will only conduct when the voltage is high enough. Since the voltage isn't symmetrical, this can mean that the tube only conducts on half of the cycle and acts as a half wave rectifier.