In this paper we present a GMRES-like iterative Krylov subspace method for solving a system of linear equations with a non-symmetric matrix and multiple right-hand sides. It minimizes residual norms and in exact arithmetic is equivalent to a GCR method. Unlike GCR, the new method requires only one basis storage. Also, as it does not use a blockwise matrix-vector product, it can be applied to sequentially accessed right-hand sides. We provide a detailed algorithm description and its template implementation in C++ with MPI and OpenMP parallelism. Several numerical experiments and a comparison with a GCR implementation are presented as well.