  An orthoset (also called an orthogonality space) is a set $X$ equipped with a
symmetric and irreflexive binary relation $\perp$, called the orthogonality
relation. In quantum physics, orthosets play a central role. In fact, a Hilbert
space gives rise to an orthoset in a canonical way and can be reconstructed
from it.
  A complex Hilbert space can be seen as a real Hilbert space endowed with a
complex structure. This fact motivates us to explore characteristic features of
real Hilbert spaces by means of the abelian groups of rotations of a plane.
Accordingly, we consider orthosets together with the groups of automorphisms
that keep the orthogonal complement of a given pair of distinct elements fixed.
We establish that, under a transitivity and a homogeneity assumption, an
orthoset arises from a projective (anisotropic) Hermitian space.
  To find conditions under which the latter's scalar division ring is $\mathbb
R$ is difficult in the present framework. However, restricting considerations
to divisible automorphisms, we can narrow down the possibilities to positive
definite quadratic spaces over an ordered field. The further requirement that
the action of these automorphisms is quasiprimitive implies that the scalar
field is a subfield of $\mathbb R$.
