The bulk crystal orientation in ice influences the flow of glaciers and ice streams. The ice c-axes fabric is most reliably derived from ice cores. Because these are sparse, the spatial and vertical distribution of the fabric in the Greenland and Antarctic ice sheets is largely unknown. In recent years, methods have been developed to determine fabric characteristics from polarimetric radar measurements. The aim of this paper is to present an improved method to infer the horizontal fabric asymmetry by precisely determining the travel-time difference using co-polarised phase-sensitive radar data. We applied this method to six radar measurements from the East Greenland Ice-core Project (EastGRIP) drill site on Greenland's largest ice stream to give a proof of concept by comparing the results with the horizontal asymmetry of the bulk crystal anisotropy derived from the ice core. This comparison shows an excellent agreement, which is a large improvement compared to previously used methods. Our approach is particularly useful for determining the vertical profile of the fabric asymmetry in higher resolution and over larger depths than was achievable with previous methods, especially in regions with strong asymmetry.