The Joacph family home and its neighbouring garage, in the Saltersgill area of Middlesbrough, were covered with messages.
These claimed that the family were terrorists, and also called for the killing of Muslims.
The Joacphs, who are practising Roman Catholics, say they are devastated by the incident.
Originally from India, they have lived in the area for three years.
Cleveland Police said such behaviour would not be tolerated, and an investigation was under way.
The council has been asked to remove the graffiti.
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