  Pseudo algebraically closed, pseudo real closed, and pseudo $p$-adically
closed fields are examples of unstable fields that share many similarities, but
have mostly been studied separately. In this text, we propose a unified
framework for studying them: the class of pseudo $T$-closed fields, where $T$
is an enriched theory of fields. These fields verify a "local-global" principle
for the existence of points on varieties with respect to models of $T$. This
approach also enables a good description of some fields equipped with multiple
$V$-topologies, particularly pseudo algebraically closed fields with a finite
number of valuations. One important result is a (model theoretic)
classification result for bounded pseudo $T$-closed fields, in particular we
show that under specific hypotheses on $T$, these fields are NTP$_2$ of finite
burden.
