Constructions of superabundant tropical curves in higher genus

We construct qualitatively new examples of superabundant tropical curves which are non-realizable in genus $3$ and $4$. These curves are in ${\mathbb{R}}^3$ and ${\mathbb{R}}^4$ respectively, and have properties resembling canonical embeddings of genus $3$ and $4$ algebraic curves. In particular, the genus $3$ example is a degree $4$ planar tropical curve, and the genus $4$ example is contained in the product of a tropical line and a tropical conic. They have excess dimension of deformation space equal to $1$. Non-realizability follows by combining this with a dimension calculation for the corresponding space of logarithmic curves.