Faster-than-Nyquist (FTN) signaling is a nonorthogonal transmission technique, which brings in intentional inter-symbol interference. This way it can significantly enhance spectral efficiency for practical pulse shapes such as the root raised cosine pulses. This paper proposes an achievable rate region for the multiple antenna (MIMO) asynchronous multiple access channel (aMAC) with FTN signaling. The scheme applies waterfilling in the spatial domain and precoding in time. Waterfilling in space provides better power allocation and precoding helps mitigate inter-symbol interference due to asynchronous transmission and FTN. The results show that the gains due to asynchronous transmission and FTN are more emphasized in MIMO aMAC than in single antenna aMAC. Moreover, FTN improves single-user rates, and asynchronous transmission improves the sum-rate, due to better inter-user interference management.