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We prove a dichotomy for o-minimal fields $\mathcal{R}$, expanded by a$T$-convex valuation ring (where $T$ is the theory of $\mathcal{R}$) and acompatible monomial group. We show that if $T$ is power bounded, then thisexpansion of $\mathcal{R}$ is model complete (assuming that $T$ is), it has adistal theory, and the definable sets are geometrically tame. On the otherhand, if $\mathcal{R}$ defines an exponential function, then the naturalnumbers are externally definable in our expansion, precluding any sort of modeltheoretic tameness.

A dichotomy for $T$-convex fields with a monomial group