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Góngora Marmolejo, Alonso de. "XXIII". Historia de Todas las Cosas que han Acaecido en el Reino de Chile y de los que lo han gobernado (1536-1575) (in Spanish). ...Llegado Jerónimo de Alderete a España en nombre de Pedro de Valdivia para negociar con su majestad, le fué necesario pasar a Ingalaterra, porque el Emperador don Carlos había renunciado todos sus reinos en el serenísimo príncipe don Felipe, su hijo, y retirado en un monasterio de religiosos, no entendía cosa alguna, ni en proveimiento de ninguna suerte; por donde le convino Alderete irse a ver con el rey, que a causa de se haber casado con la reina de Ingalaterra estaba en aquel reino.
The Emperor has renounced ALL HIS KINGDOMS and retired to a monastery, ie, Yuste. So, nothing to do with Felipe being King of Chile, but King of Spain and the Indies.
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I have two more sources both of which confirm that the sovereignty of Chile was ceded to Philip by his father. They refer to Chile as Reino de Chile.
The conquistador Pedro de Valdivia sent Geromino Alderete at the end of 1552 to Spain to confirm his governorship of Chile. However, on arrival in Madrid he found out that Carlos I (Charles V) had transferred sovereignty of Chile to his son, Philip.
As a result, Alderete had to travel to London for an audience with Philip at the court of Mary I. Philip granted Valdivia the governorship of Chile for life. (He could not have done this if he was not sovereign of Chile.)
On the journey back to Spain, Alderete received news that Valdivia had died. So, he returned to London where Philip made him governor.
These and other accounts make it clear that Philip enjoyed sovereignty of Chile. With that must have come some sort of title. The wording Reino de Chile is relatively common in Chilean history.
Sources:
Diccionario Biográfico Colonial de Chile, José Toribio Medina
Historia de Todas las Cosas que han Acaecido en el Reino de Chile y de los que lo han gobernado (1536-1575), Alonso de Góngora Marmolejo