We examine in this paper the problem of anonymizing the provenance of collection-oriented workflows, in which the constituent modules use and generate sets of data records. Despite their popularity, this kind of workflows has been overlooked in the literature wrt privacy. We, therefore, set out in this paper to examine the following questions: How the provenance of a collection-based module can be anonymized? Can lineage information be preserved? Beyond a single module, how can the provenance of a whole workflow be anonymized? As well as addressing the above questions, we report on evaluation exercises that assess the effectiveness and efficiency of our solution. In particular, we tease apart the parameters that impact the quality of the obtained anonymized provenance information.