Just to add, I know about one specific subject at the TRNC bank market – Turkish Bank Group. Native at TRNC (established 1901), having member banks at TRNC, Turkey and UK as well – and having as a member Allied Turkish Bank IBU Ltd <=> dedicated offshore bank. Seems interesting; but honestly – I'd be afraid of political instability (in the long-term perspective) of TRNC.
And just for the record, there is also a branch of Halkbank at TRNC. (Halkbank is at Northern Macedonia, too; and AFAIK they are open to non-residents.)︀