The Garlean flagship's "base" as you refer to it, is Ala Mhigo and the various imperial castra in Gyr Abania. The one the readily comes to mind is Castellum Velodyna whose huge tower was originally meant for mooring airships. In addition, the Garleans were said to have built a bunch of manufacturing factories all over Gyr Abania. The flagship itself would probably have been built back in Garlemald where most of the imperial engineers who do design work live.