Kodaira embedding theorem can be regarded as a vast generalizaton of the projectivity criterion for complex tori: indeed, the Riemann conditions essentially say that the line bundle defined by the polarisation is positive.

If $k$ is a complete field endowed with non-Archimedean absolute value, then Riemann conditions for analytic tori $\mathbb{G}_m^n(k)/\mathbb{Z}^n$ can also be given, polarisation becomes a map $\varphi: \mathbb{Z}^n \to \mathrm{Hom}(\mathbb{G}_m^n, \mathbb{G}_m)$ such that $\varphi(\lambda)(\lambda')=\varphi(\lambda')(\lambda)$ for any $\lambda, \lambda' \in \mathbb{Z}^n$, and the positivity condition now reads as that the form $\sigma(\lambda, \lambda')=-\log |\varphi(\lambda)(\lambda')|$ is positive definite (all this is nicely exposed in Fresnel and van der Put's book "Rigid analytic geometry and its applications", Section 6.5).

It is then natural to ask: does there exist a criterion of projectivity akin to Kodaira embedding theorem for rigid analytic varieties that are not tori?