  The aim of this article is to apply a Floer theory to study symmetric
periodic Reeb orbits. We define positive equivariant wrapped Floer homology
using a (anti-)symplectic involution on a Liouville domain and investigate its
algebraic properties. By a careful analysis of index iterations, we obtain a
non-trivial lower bound on the minimal number of geometrically distinct
symmetric periodic Reeb orbits on a certain class of real contact manifolds.
This includes non-degenerate real dynamically convex starshaped hypersurfaces
in $\mathbb{R}^{2n}$ which are invariant under complex conjugation. As a
result, we give a partial answer to the Seifert conjecture on brake orbits in
the contact setting.
