In this paper, a high performance MAP receiver with significantly reduced complexity is proposed for multiple-access channel with M - ary quasi orthogonal signaling. The proposed architecture operates on the channel and code matched filters (MF) belonging to each users signaling waveforms. The proposed MAP receiver is based on the factor graph (FG) obtained by using the equivalent channel model, and it operates on forward and backward directions along signaling intervals iteratively. Also, it is substantiated with bidirectional decision feedback (BDF) in order to limit the number of states used by each user. This Ungerboeck type MAP receiver, where Sum-Product Algorithm (SPA) with the support of RSSE, without need for whitening filter shows a very close performance to matched filter bound (MFB) at significantly reduced complexity for multipath channels with large delay spread and number of users.