Kalitta Air. Practical to live overseas, but I don't think they have Home Basing overseas, exactly.
ATI
NetJets has Gateway Basing I think..pretty close.
Atlas AIr has Gateway Travel...not as good as Home Basing, but they buy you a ticket to/from work, and it's practical to live overseas.
Travel Mgmt (TMC) has home basing, but I wouldn't recommend them.