“Registration of stay” means officially registering your address and presence in Croatia with the competent authority (usually the local police station / Ministry of Interior office).
This is not a HomeInZagreb rule and not a landlord rule – it comes from Croatian and EU laws on:
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residence and stay of foreigners, and
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obligations of people living in a country for a certain period of time.
If you stay in Croatia without being properly registered or without the correct visa/residence status, you may face legal or immigration problems, regardless of your rental contract.