Diffusion MRI allows the structural connectivity of the whole brain (the 'tractogram') to be estimated in vivo non-invasively using streamline tractography. The biological accuracy of these data sets is however limited by the inherent biases associated with the reconstruction method. Here we propose a method to retrospectively improve the accuracy of these reconstructions, by selectively filtering out streamlines from the tractogram in a manner that improves the fit between the streamline reconstruction and the underlying diffusion images. This filtering is guided by the results of spherical deconvolution of the diffusion signal, hence the acronym SIFT: spherical-deconvolution informed filtering of tractograms. Data sets processed by this algorithm show a marked reduction in known reconstruction biases, and improved biological plausibility. Emerging methods in diffusion MRI, particularly those that aim to characterise and compare the structural connectivity of the brain, should benefit from the improved accuracy of the reconstruction.