  Assuming a certain form of resolution of singularities, we prove a general
existential Ax-Kochen/Ershov principle for tamely ramified fields in all
characteristics. This specializes to well-known results in residue
characteristic $0$ and unramified mixed characteristic. It also encompasses the
conditional existential decidability results known for $\mathbb{F}_p(\!(t)\!)$
and its finite extensions, due to Denef-Schoutens. On the other hand, it also
applies to the setting of infinite ramification, providing us with an abundance
of infinitely ramified extensions of $\mathbb{Q}_p$ and $\mathbb{F}_p(\!(t)\!)$
that are existentially decidable.
