Recent publications have explored coherent radar detection in a compound Gaussian clutter environment with inverse gamma texture, since the latter clutter model has been validated for X-band high-resolution maritime surveillance radar clutter returns. This paper explores the development of coherent constant false alarm rate (CFAR) detectors for this scenario. In the first instance, a detector is constructed with explicit knowledge of the clutter parameters. It is then shown that the probability of false alarm/threshold relationship does not vary with the clutter power. To achieve a CFAR detector, clutter parameter approximations are then introduced, and the cost associated with this is then analysed.