  We classify maximal quartic generalised projective special real curves up to
equivalence. A maximal quartic generalised projective special real curve
consists of connected components of the intersection of the hyperbolic points
of a quartic homogeneous real polynomial $h:\mathbb{R}^2\to\mathbb{R}$ and its
level set $\{h=1\}$. Two such curves are called equivalent if they are related
by a linear coordinate transformation. As an application of our results we
prove that quartic generalised projective special real manifolds that are
homogeneous spaces have non-regular boundary behaviour, meaning that the
differential of each of these spaces' defining polynomials vanishes identically
on a ray in the boundary of the cone spanned by the corresponding manifold.
Lastly we describe the asymptotic behaviour of each curve.
