An achievable rate region for the asynchronous multiple access channel with feedback is established through the use of superposition coding, list decoding and time sharing. The calculation results demonstrate that lack of synchronization does not affect the achievable rate region when the code block length tends to infinity, and that if the length of the code word is finite, especially not sufficiently larger than a fixed maximal delay, the asynchronization will cause a loss of the rate region. The amount of such a loss with its explanation for the reason is given, and the difference between the losses for the asynchronous multiple access channel with and without feedback is also discussed in this paper.