Connectivity hyperalignment can be used to estimate a single shared response space across disjoint datasets. We develop a connectivity-based shared response model that factorizes aggregated fMRI datasets into a single reduced-dimension shared connectivity space and subject-specific topographic transformations. These transformations resolve idiosyncratic functional topographies and can be used to project response time series into shared space. We evaluate this algorithm on a large collection of heterogeneous, naturalistic fMRI datasets acquired while subjects listened to spoken stories. Projecting subject data into shared space dramatically improves between-subject story time-segment classification and increases the dimensionality of shared information across subjects. This improvement generalizes to subjects and stories excluded when estimating the shared space. We demonstrate that estimating a simple semantic encoding model in shared space improves between-subject forward encoding and inverted encoding model performance. The shared space estimated across all datasets is distinct from the shared space derived from any particular constituent dataset; the algorithm leverages shared connectivity to yield a consensus shared space conjoining diverse story stimul