Diffusion-weighted imaging and tractography provide a unique, non-invasive technique to study the macroscopic structure and connectivity of brain white matter in vivo. Global tractography methods aim at reconstructing the full-brain fiber configuration that best explains the measured data, based on a generative signal model. In this work, we incorporate a multi-shell multi-tissue model based on spherical convolution, into a global tractography framework, which allows to deal with partial volume effects. The required tissue response functions can be estimated from and hence calibrated to the data. The resulting track reconstruction is quantitatively related to the apparent fiber density in the data. In addition, the fiber orientation distribution for white matter and the volume fractions of gray matter and cerebrospinal fluid are produced as ancillary results. Validation results on simulated data demonstrate that this data-driven approach improves over state-of-the-art streamline and global tracking methods, particularly in the valid connection rate. Results in human brain data correspond to known white matter anatomy and show improved modeling of partial voluming. This work is an important step toward detecting and quantifying white matter changes and connectivity in healthy subjects and patients.