  We study the geometry of tropical extensions of hyperfields, including the
ordinary, signed and complex tropical hyperfields. We introduce the framework
of `enriched valuations' as hyperfield homomorphisms to tropical extensions,
and show that a notable family of them are relatively algebraically closed. Our
main results are hyperfield analogues of Kapranov's theorem and the Fundamental
theorem of tropical geometry. Utilizing these theorems, we introduce fine
tropical varieties and exhibit a number of applications to motivate their
study.
