  If $C$ is a curve over $\mathbb{Q}$ with genus at least $2$ and
$C(\mathbb{Q})$ is empty, then the class of fields $K$ of characteristic 0 such
that $C(K) = \varnothing$ has a model companion, which we call $C\mathrm{XF}$.
The theory $C\mathrm{XF}$ is not complete, but we characterize the completions.
Using $C\mathrm{XF}$, we produce examples of fields with interesting
combinations of properties. For example, we produce (1) a model-complete field
with unbounded Galois group, (2) an infinite field with a decidable first-order
theory that is not ``large'' in the sense of Pop, (3) a field that is
algebraically bounded but not ``very slim'' in the sense of Junker and
Koenigsmann, and (4) a pure field that is strictly NSOP$_4$, i.e., NSOP$_4$ but
not NSOP$_3$. Lastly, we give a new construction of fields that are virtually
large but not large.
