  Tropical toric varieties are partial compactifications of finite dimensional
real vector spaces associated with rational polyhedral fans. We introduce
plurisubharmonic functions and a Bedford--Taylor product for Lagerberg currents
on open subsets of a tropical toric variety. The resulting tropical toric
pluripotential theory provides the link to give a canonical correspondence
between complex and non-archimedean pluripotential theories of invariant
plurisubharmonic functions on toric varieties. We will apply this
correspondence to solve invariant non-archimedean Monge--Amp\`ere equations on
toric and abelian varieties over arbitrary non-archimedean fields.
