Motivated by the recent interest in approximate message passing (AMP) for matrix-valued linear observations with superposition of \emph{multiple statistically asymmetric signal sources}, we introduce a multi-source AMP framework in which the dictionary matrices associated with each signal source are drawn from a \emph{random semi-unitary ensemble} (rather than the standard Gaussian matrix ensemble.) While a similar model has been explored by Vehkaper{ä}, Kabashima, and Chatterjee (2016) using the replica method, here we present an AMP algorithm and provide a high-dimensional yet \emph{finite-sample} analysis. As a proof of concept, we show the effectiveness of the proposed approach on the problem of \emph{message detection and channel estimation} in an unsourced random access scenario in wireless communication.