  For any stable curve $f$ in an exploded manifold, this paper constructs a
family of curves $\hat f$ with universal tropical structure which contains $f$.
Such a family has the property that any other family of curves containing $f$
is locally a small modification of a family which factors through $\hat f$. As
such, families of curves with universal tropical structure play an important
role in the analysis of the moduli stack of curves and the construction of
Gromov-Witten invariants of exploded manifolds.
