We design an explicit low-complexity coding scheme that achieves the multiple access channel resolvability region for an arbitrary discrete memoryless multiple access channel whose input alphabets have prime cardinalities. Unlike previous works, we do not assume channel symmetry and rely on rate-splitting to avoid time sharing when it is known to be unnecessary. The idea of our construction is to reduce the problem of multiple access channel resolvability to a combination of several source resolvability problems. Our coding scheme relies on polar codes for source coding to implement source resolvability, and a block Markov coding scheme that performs randomness recycling in the different encoding blocks.