The following setting would be responsible for updating Lists in conjunction with Data Extensions that are used for Triggered sends:
Here is the page that this is contained in within the documentation:
From a holistic perspective, maintaining Data in such a way is not ideal. In a perfect world, you would want to use a single source of truth for attribute data (Data Extension/List Fields). Specifically meaning that all things that referenced said attribute data were all aligned to use List Data or conversely all aligned to use Data Extension data.
Attempting to have data be 100% in sync with two objects (Lists/Data Extensions) that are mutually exclusive inevitably presents problems. Specifically, as there is no auto-sync between Lists and Data Extensions, there will eventually be a case where at some point where data between the two objects is not in sync for some period of time. This period of time can be minimized with processes set to match the data up but will still exist. It is also a possibility that data can come from an unexpected place/misunderstood process that updates one set of data but not the other within more complex accounts that have more moving parts.
In most situations where this comes up there is a desire for tracking data which is easily discoverable/available for Lists but less so for Data Extensions. This can still be obtained for Data Extensions however would just require marrying said Data upon the "Sendable Relationship" or "Primary Key" of record with tracking data.