Brian Blessed as the Ghost in Kenneth Branagh's Hamlet movie - and whispering, not shouting!
In the preface to Lyrical Ballads by Wordsworth and Coleridge, the 'romantic' was given a definition in terms of Wordsworth focusing on the numinous qualities of Nature, and Coleridge upon the Supernatural. So the Fantasy aspect of Romanticism - such as we see in The Inklings - was present from the start.
Shakespeare had a role in Romanticism, in that his renewed popularity came from a German reappraisal (probably originating with Herder, who worked from translated excerpts of Shakespeare - and greatly amplified in England by the Shakespeare lectures of Coleridge) that saw Shakespeare in such terms; as a 'wild', spontaneous, natural genius.
Shakespeare was often linked with the then extremely high valuation of the strange, semi-faked 'poems' of 'Ossian' - which were a 'translation', plus very extensive reworking and expansion, of Gaelic songs and stories collected in the highlands and islands of Scotland by James Macpherson and published in the middle 1700s. So that Homer, Ossian and Shakespeare were found linked in a 'bardic' lineage - for instance, this is later referenced by Emerson and Thoreau.
Shakespeare came from a 'recusant' Catholic, and therefore pre-Reformation, sensibility on his mother's side. She was Mary Arden, and members of the Arden family were part of a concentration of families trying to practice the Old religion; Ardens even involved in, and some executed for, spying, rebellion, and assassination plots.
So, Shakespeare had an interest in magic, fairies, ghosts, witches and the like; that can be seen from A Midsummer Night's Dream and The Tempest (the only plays whose plots were essentially original) and others of his greatest work such as Macbeth and Hamlet. And, in general, many of the plays have a strong 'premodern' element of fate or providence governing the affairs of Men. However, as Tolkien noted with disappointment, such elements are seldom given full value or treated with full seriousness - it is mostly a matter of hints and epiphanies.
With the dawning of the Romantic consciousness, Shakespeare was therefore ripe for revival and reinterpretation in this light - despite that in other respects, his work was often very 'artificial', ironic, fashionable, or simply derivative (in terms of basic plots and subject matter).
Shakespeare can therefore be seen as spanning from the Medieval consciousness, which simply took magic and the supernatural for granted as a real and objective part of the external environment; the early modern consciousness, which treated such matters ironically and satirically - as being unreal and delusional - purely a part of subjective consciousness; and the Romantic consciousness which (in its highest development; e.g. Coleridge, Novalis, Blake) returns to the pre-modern in conscious thought, with a realism that includes as necessary, both the subjective and the objective.
In other words, the Romantic concept of the magical-supernatural is that these things are real; and/but - like all real things - their reality is neither wholly in the external environment nor the internal consciousness, but in the necessary apprehension of the environment by consciousness. They (both the supernatural and all real phenomena) are indeed out-there, and/but they require the subjective consciousness to be known; and only the known is real. (Only the known is part of Creation - all else is unknowable chaos.)
The magical is real, but reality has a new location, in the realm of thinking. Neither purely out-there nor in-here, but in a new conception of the potential of the world of thinking to know directly and perhaps participate in divine creation. This is the thinking of the 'creative genius', who is seen by Romantics as (potentially) the proper mode of all Men.
So the ghost in Hamlet is 'really there' but requires someone to see and hear it; if there was no consciousness present, there would be no ghost.