If you go by the letter of the law, any offshore company controlled by a resident in Cyprus is tax resident in Cyprus.
In reality, no one is bothering non-domiciled residents (foreigners) who don't pay local tax on their offshore companies. There are many reasons why this is or may be, but for the moment, it's generally considered safe but probably not entirely legal to live in Cyprus, have an offshore company, and not pay any tax. A lot of larger, international businesses run parallel entities: one in Cyprus which pays local salaries to employees and is used to obtain work permits for non-EU citizens and another, offshore which pays additional salaries under the table and pays no corporate tax.
However, more and more are opting to form a local company in Cyprus. As a non-domiciled resident, if you own a local Cypriot company, you can normally get away with paying the 12.50% corporate tax and then pay yourself tax free dividends. There are some caveats to consider so make sure you work with a qualified adviser but that's the general gist of it.