Spurred by the recent proposal of Braginskii and co-workers, we study the possibility of making quantum nonperturbing (qnp) measurements of coherent states of the electromagnetic field. As a model problem we consider the Compton scattering of electrons from a coherent photon beam. We find (to second order) that the electrons can scatter with only a small probability (per scattered electron) of disturbing the coherence of the ongoing incident beam. Measurement of the scattering rate for a known density of target electrons gives direct information about the absolute square of the coherence eigenvalue of the incident phloton beam. We also consider the (somewhat academic) process of pair production by colliding coherent photon beams. In this case coherence is completely preserved at second ord.er, and only higher-order amplitudes (e.g., Delbrtick scattering) contribute to its disturbance. Our results, though not directly related to Braginskii's proposed experiment, suggest the potential importance of coherent states for the design of a quantum nonperturbing measuring device.