We show that gravitational-wave signals from compact binary mergers may be better distinguished from instrumental noise transients by using Bayesian models that look for signal coherence across a detector network. This can be achieved even when the signal power is below the usual threshold for detection. This method could reject the vast majority of noise transients, and therefore increase sensitivity to weak gravitational waves. We demonstrate this using simulated signals, as well as data for GW150914 and LVT151012. Finally, we explore ways of incorporating our method into existing Advanced LIGO and Virgo searches to make them significantly more powerful.