The double pulsar system exhibits unique and spectacular eclipses when pulsar A passes behind pulsar B. We present 3.5 years of eclipse monitoring which provides an unprecedented way of probing pulsar magnetospheres, geometric orientation as well as testing general relativity The eclipse light curve of pulsar A has a rich phenomenology. This includes little observed radio frequency dependence and periodic flux modulations in phase with the rotation of pulsar B, which allow the flux to reach uneclipsed levels in narrow windows. Over the monitoring period, the eclipse profile noticeably changed and we observe a modification in the modulation behavior. We quantitatively analyzed the data using the magnetospheric synchrotron absorption eclipse model proposed by Lyutikov and Thompson under the assumption of a simple dipolar magnetic field geometry The success of the model at reproducing the eclipse profile indicates that the magnetic field configuration is mainly dipolar at a distance of about 0.1 light‐cylind...