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This paper addresses the challenge of leveraging multiple embedding spaces for multi-shop personalization, proving that zero-shot inference is possible by transferring shopping intent from one website to another without manual intervention. We detail a machine learning pipeline to train and optimize embeddings within shops first, and support the quantitative findings with additional qualitative insights. We then turn to the harder task of using learned embeddings across shops: if products from different shops live in the same vector space, user intent - as represented by regions in this space - can then be transferred in a zero-shot fashion across websites. We propose and benchmark unsupervised and supervised methods to "travel" between embedding spaces, each with its own assumptions on data quantity and quality. We show that zero-shot personalization is indeed possible at scale by testing the shared embedding space with two downstream tasks, event prediction and type-ahead suggestions. Finally, we curate a cross-shop anonymized embeddings dataset to foster an inclusive discussion of this important business scenario.