Plato, in a spurious sense, is often regarded as a ‘realist’ since he assigned ‘real’ status to his ‘forms’. However, he is a proponent of an idealist’s perspective because ideas (forms), though projected as belonging to an independent realm, but actually existed within mind. For Plato, forms constituted objective reality and the ‘observed reality’ was nothing but shadow imagery of those ‘forms’. If we regard Plato as a realist then we would be treating idealism as realism because a treatment of mind contents as an independent realm is a sign of extreme idealism and not a mark of realism.
In the Platonic theory, external things are shadow images and real things are basically just ideas of mind but anyhow Plato regards them absolute forms that exist in a supposed objective and real ‘world of ideas’ and not within mind. Perhaps it is his insistence that forms belong to an objective ‘world of ideas’ and not to the mind that he is often acknowledged as a realist which he is not actually because basically he just assigned status of an independent realm to forms which are nothing but ideas that belong to mind and do not possibly exist anywhere else.