How does a database conciliate data to files on distributed filesystems?
In simple terms, the DBMS doesn't care (or in some cases even know) if the file system where its files are located is distributed or not. In a high availability scenario with shared (distributed) storage an active-passive setup is most commonly used, where only one instance of the DBMS at a time is running and has access to the database files. It's the responsibility of the cluster manager (software or human) to control DBMS instance startup and failover.

In a shared data cluster, such as Oracle RAC or Db2 pureScale, there are multiple instances of the DBMS running concurrently and accessing the same set of database files, and a special component of the cluster manages concurrent access to the files by the DBMS processes.

That "special component" is usually tightly integrated with the DBMS itself but the actual implementations differ wildly. Oracle Clusterware plays this role in an Oracle RAC; Cluster Cache Facility in Db2 pureScale; both are tightly integrated with their respective underlying shared file systems.

I'm not aware of any open source shared-data cluster databases; they all tend to choose shared-nothing architectures, which are somewhat simpler to implement and more tolerant towards low-end hardware.