Import/Link database from raw MySQL database file

On my computer, I have 2 linux systems I use to work: a Fedora and a Ubuntu. Each of them have their own MySQL instance. But some projects needs to have a comon DB on both systems. So, here is my need: I'd like to "link" the raw MySQL file from /var/lib/mysql to my other OS, so any edit in this database from one OS will be also present on the same database on the other OS. Is it possible? How? I tried to link directly the file, but as I expected, it does not work... I guess this is related with the information_schema that was not modified as it should when normally creating a DB.

Thank you for your help

Edit: both systems are on the same machine, so I can't connect to other OS MySQL instance: I can't boot on both OS at the same time