  We launch the study of the tropicalization of the symplectic Grassmannian,
that is, the space of all linear subspaces that are isotropic with respect to a
fixed symplectic form. We formulate tropical analogues of several equivalent
characterizations of the symplectic Grassmannian and determine all implications
between them. In the process, we show that the Pl\"ucker and symplectic
relations form a tropical basis if and only if the rank is at most 2. We
provide plenty of examples that show that several features of the symplectic
Grassmannian do not hold after tropicalizing. We show exactly when do conormal
fans of matroids satisfy these characterizations, as well as doing the same for
a non-constant coefficient generalization. Finally, we propose several
directions to extend the study of the tropical symplectic Grassmannian.
