What would be the classification of banks (which deal in conventional interest based products) in Gulf Countries eg. UAE, Oman, Qatar, if the shareholders are non Shia Muslims or organisations whose shareholders are non Shia individuals. Would they be classified differently from banks in kafir / democratic nations

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If these banks are 100% owned by Muslim people, they are considered private banks, from which taking loans is not permissible. However, you can deposit money in them without asking for profit, or considering the profit announced by the bank as your right. With these two coditions, you can take the given profit.