Quantum state reconstruction of weak propagating microwaves to date requires the use of linear amplifiers. We introduce a theory which, even in the presence of significant amplifier noise, allows one to use these devices for measuring all quadrature moments of propagating quantum microwaves based on cross correlations from a dual-path amplification setup. Simultaneously, the detector noise properties are determined, allowing for tomography. We demonstrate the feasibility of our novel concept by proof-of-principle experiments with classical mixtures of weak coherent microwaves.