Hilbertian kernel methods and their positive semidefinite kernels have been extensively used in various fields of applied mathematics and machine learning, owing to their several equivalent characterizations. We here unveil an analogy with concepts from tropical geometry, proving that tropical positive semidefinite kernels are also endowed with equivalent viewpoints, stemming from Fenchel-Moreau conjugations. This tropical analogue of Aronszajn's theorem shows that these kernels correspond to a feature map, define monotonous operators, and generate max-plus function spaces endowed with a reproducing property. They furthermore include all the Hilbertian kernels classically studied as well as Monge arrays. However, two relevant notions of tropical reproducing kernels must be distinguished, based either on linear or sesquilinear interpretations. The sesquilinear interpretation is the most expressive one, since reproducing spaces then encompass classical max-plus spaces, such as those of (semi)convex functions. In contrast, in the linear interpretation, the reproducing kernels are characterized by a restrictive condition, von Neumann regularity. Finally, we provide a tropical analogue of the "representer theorems", showing that a class of infinite dimensional regression and interpolation problems admit solutions lying in finite dimensional spaces. We illustrate this theorem by an application to optimal control, in which tropical kernels allow one to represent the value function.