We propose a class of projected Krylov methods for the solution of unsymmetric augmented systems of equations such as those arising from the finite-element formulation of Navier-Stokes multi-fluid flow problems. The iterative methods only rely on matrix-vector products with the (1,1) block of the augmented matrix - not with its transpose - and on a one-time symmetric indefinite factorization of a projection matrix. No computation of Schur complements or generalized inverses is necessary, nor is the computation of a nullspace or of a range-space basis. Numerical results coming from fluid dynamics examples illustrate the present approach and compare it to a direct application of a Krylov method to the augmented system, and to a direct factorization of the system - assuming that the latter is feasible.