The pure-interference bosonic multiple access channel has two senders and one receiver, such that the senders each communicate with multiple temporal modes of a single spatial mode of light. The channel mixes the input modes from the two users pairwise on a lossless beamsplitter, and the receiver has access to one of the two output ports. In prior work, Yen and Shapiro found the capacity region of this channel if encodings consist of coherent-state preparations. Here, we demonstrate how to achieve the coherent-state Yen-Shapiro region (for a range of parameters) using a sequential decoding strategy, and we show that our strategy outperforms the rate regions achievable using conventional receivers. Our receiver performs binary-outcome quantum measurements for every codeword pair in the senders' codebooks. A crucial component of this scheme is a non-destructive "vacuum-or-not" measurement that projects an n-symbol modulated codeword onto the n-fold vacuum state or its orthogonal complement, such that the post-measurement state is either the n-fold vacuum or has the vacuum removed from the support of the n symbols' joint quantum state. This receiver requires the additional ability to perform multimode optical phase-space displacements which are realizable using a beamsplitter and a laser.