Let $V$ a real vector space, endowed with an hermitian form.
To set things more formally, let $(e_1, \dotsc, e_n)$ a basis of $V$, then the data of the hermitian form is encoded in an Hermitian matrix $S=(s_{i,j})$, defining by linearity a form: $$ \langle x, y \rangle = \sum_{i,j} s_{i,j} x_i y_j $$ where $x = \sum_i x_ie_i$ and $y = \sum_i y_ie_i$ generic elements of $V$. Then $V\otimes_\mathbb{R} \mathbb{C}$ can be trivially viewed as an hermitian vector space through the form defined by $S$. But I'm interested only in the real vector space $V$ and not its complexification.
Is there a known theory for the study of such spaces?