Constructions of superabundant tropical curves in higher genus

We construct qualitatively new examples of superabundant tropical curves which are non-realisable in genuses 3 and 4. These curves are in $R^3$ and $R^4$, 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-realisability follows by combining this with a dimension calculation for the corresponding space of logarithmic curves.