Two main problems arise in the Multiple Access Channel (MAC): interference from different users, and additive noise channel noise. Maximum A-Posteriori (MAP) joint decoding or successive interference cancellation are known to be capacity-achieving for the MAC when paired with appropriate codes. We extend the recently proposed Soft Guessing Random Additive Noise Decoder (SGRAND) to guess, using soft information, the effect of noise on the sum of users' transmitted codewords. Next, we manage interference by applying ZigZag decoding over the resulting putative noiseless MAC to obtain candidate codewords. Guessing continues until the candidate codewords thus obtained pertain to the corresponding users' codebooks. This MAC SGRAND decoder is a MAP decoder that requires no coordination between users, who can use arbitrary moderate redundancy short length codes of different types and rates.