We prove that every toric monoid appears in a space of maps from tropical curves to an orthant. It follows that spaces of logarithmic maps to Artin fans exhibit arbitrary toric singularities: a virtual universality theorem for logarithmic maps to pairs. The target rank depends on the chosen singularity:
we show that the cone over the 7-gon never appears in a space of maps to a rank 1 target. We obtain similar results for tropical maps to affine space.

Comment: 22 pages. Comments welcome. v4: Published version in journal style

• 14A21 - Logarithmic algebraic geometry, log schemes
• 14H10 - Families, moduli of curves (algebraic)
• 14M25 - Toric varieties, Newton polyhedra, Okounkov bodies
• 14N35 - Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
• 14T20 - Geometric aspects of tropical varieties