A common tool to process and interpret multimodal data is to represent the data in a sparse Tucker format, decomposed as a sparse core tensor and dictionary matrices for each modal dimension. In real-world applications one may be presented with a composition of several tensors, each with its own sparse Tucker representation and collection of dictionaries. The Tucker model and associated recovery algorithms struggle to accurately separate composite tensors in this situation, either having difficulty with the overcomplete dictionaries or not fully taking advantage of the special structure of the decomposition. To address these deficiencies, we introduce an overcomplete sparse Tucker model and an iterative algorithm to separate a composite sparse Tucker tensor. The method, which is based on soft-thresholding shrinkage techniques, is demonstrated to effectively separate overcomplete tensors and recover the sparse component tensors on real-world datasets, and to do so more accurately than other Tucker methods.