  We describe a natural open stratum in the moduli space of smooth real pointed
quartic curves in the projective plane. This stratum consists of real
isomorphism classes of pairs $(C, p)$ with $p$ a real point on the curve $C$
such that the tangent line at $p$ intersects the curve in two distinct points
besides $p$. We will prove that this stratum consists of $20$ connected
components. Each of these components has a real toric structure defined by an
involution in the Weyl group of type $E_7$.
