Mostly it comes down to definitions for FTL purposes, under EASA anyway.
"ORO.FTL.105 Definitions
(14) “home base” means the location, assigned by the operator to the crew member, from where the crew member normally starts and ends a duty period or a series of duty periods and where, under normal circumstances, the operator is not responsible for the accommodation of the crew member concerned;"
"ORO.FTL.200 Home base
An operator shall assign a home base to each crew member."
"ORO.FTL.245 Records of home base, flight times, duty and rest periods
(a) An operator shall maintain, for a period of 24 months:
(1) individual records for each crew member including:
(iv) assigned home base;"
With all that in mind, whilst not "illegal" I believe you would have a very difficult time convincing any Authority of the concept of dual basing. There's no definition, but if you changed the base every week I think they wouldn't look favourably, each month maybe, six monthly more so. Probably a test of 'reasonableness' from the Authority's perspective on this: "where, under normal circumstances, the operator is not responsible for the accommodation of the crew member concerned".