What properties a matrix need to satisfy to form a real vector space

I know the definition of vector space (i.e. it should be closed under addition and multiplication). I read in a book (‘Linear Algebra done right’ by Sheldon Axler) that the numbers(complex or real) inside the matrices decides wether the vector space they form will be complex or real.

But my question is, since hermitian matrices have complex numbers inside it, then how can it form a real vector space.

A same question is asked here Show that Hermitian Matrices form a Vector Space

But in the answer, he tells that hermitian matrices are not closed under multiplication, how does it implies that it forms real vector vector space (to this logic, hermitian matrices must not even form a vector space, since they are sometimes complex, and on multiplication by complex number it becomes real?