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As the article explains, the palace name arises from the fact that it was an almunia or real (from Arabic riyad, garden), not because it was a royal residence.
So, in this case, real≠royal. That's why the name of the palace in Spanish is Palacio del Real and not Palacio Real, or in Valencian Palau del Real instead of Palau Reial.
The same would happen with Jardines del Real (Real Gardens), almáciga del Real (Real plantation), pont del Real (Real Bridge), Pla del Real, El Real de Gandia, ...