We introduce an iterative method named GPMR for solving 2X2 block unsymmetric linear systems. GPMR is based on a new process that reduces simultaneously two rectangular matrices to upper Hessenberg form and that is closely related to the block-Arnoldi process. GPMR is tantamount to BLOCK-GMRES with two right-hand sides in which the two approximate solutions are summed at each iteration, but requires less storage and work per iteration. We compare the performance of GPMR with GMRES and BLOCK-GMRES on linear systems from the SuiteSparse Matrix Collection. In our experiments, GPMR terminates significantly earlier than GMRES on a residual-based stopping condition with an improvement ranging from around 10% up to 50% in terms of number of iterations. We also illustrate by experiment that GPMR appears more resilient to loss of orthogonality than BLOCK-GMRES.