As cloud services become central in an increasing number of applications, they process and store more personal and business-critical data. At the same time, privacy and compliance regulations such as GDPR, the EU ePrivacy regulation, PCI, and the upcoming EU Cybersecurity Act raise the bar for secure processing and traceability of critical data. Especially the demand to provide information about existing data records of an individual and the ability to delete them on demand is central in privacy regulations. Common to these requirements is that cloud providers must be able to track data as it flows across the different services to ensure that it never moves outside of the legitimate realm, and it is known at all times where a specific copy of a record that belongs to a specific individual or business process is located. However, current cloud architectures do neither provide the means to holistically track data flows across different services nor to enforce policies on data flows. In this paper, we point out the deficits in the data flow tracking functionalities of major cloud providers by means of a set of practical experiments. We then generalize from these experiments introducing a generic architecture that aims at solving the problem of cloud-wide data flow tracking and show how it can be built in a Kubernetes-based prototype implementation.