In light of the two leading candidates for the title of "tropical variety", we need to distinguish two competing (cooperating?) notions of tropical conics: 1) bend loci defined by quadratic tropical polynomials; and 2) congruence varieties defined by quadratic tropical equations.

Quadratic bend varieties

The bendy option for tropical conics is the bend locus of a quadratic tropical polynomial. Focusing on plane conics, these are the bend loci of polynomials of the form $f(x,y)=(a\odot x^{\odot 2})\oplus (b\odot x\odot y)\oplus (c\odot y^{\odot 2})\oplus (d\odot x)\oplus (e\odot y)\oplus f$, where $a,b,c,d,e,f\in \mathbf{T}$ (and at least one of $a,b,c$ is not equal to $0_{\mathbf{T}}=-\mathrm{\infty }$.) It turns out that these conics fall into exactly 20 combinatorial types, depending on the relations between the coefficients.

(INSERT EXAMPLES OF EVERY TYPE, ALONG WITH AARON'S CONDITIONS FOR EACH TYPE)

Quadratic congruence varieties

The congruence option for tropical conics is the congruence variety of a quadratic tropical equation, i.e., an equation of the form $f(x,y)=g(x,y)$, involving two tropical polynomials of degree at most two (and at least one term of degree two). These congruence varieties share many similar features with the bend loci above, but also have the unique possibility of including some two-dimensional regions in the plane.

(INSERT A BUNCH OF EXAMPLES)