Browser fingerprinting is a technique that collects information about the browser configuration and the environment in which it is running. This information is so diverse that it can partially or totally identify users online. Over time, several countermeasures have emerged to mitigate tracking through browser fingerprinting. However, these measures do not offer full coverage in terms of privacy protection, as some of them may introduce inconsistencies or unusual behaviors, making these users stand out from the rest. We address these limitations by proposing a novel approach that minimizes both the identifiability of users and the required changes to browser configuration. To this end, we exploit clustering algorithms to identify the devices that are prone to share the same or similar fingerprints and to provide them with a new non-unique fingerprint. We then use this fingerprint to automatically assemble and run web browsers through virtualization within a docker container. Thus all the devices in the same cluster will end up running a web browser with an indistinguishable and consistent fingerprint.