  Let $\mathcal{W}\subset\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ be a
surface given by the vanishing of a $(2,2,2)$-form. These surfaces admit three
involutions coming from the three projections
$\mathcal{W}\to\mathbb{P}^1\times\mathbb{P}^1$, so we call them
$\textit{tri-involutive K3 (TIK3) surfaces}$. By analogy with the classical
Markoff equation, we say that $\mathcal{W}$ is of $\textit{Markoff type (MK3)}$
if it is symmetric in its three coordinates and invariant under double sign
changes. An MK3 surface admits a group of automorphisms $\mathcal{G}$ generated
by the three involutions, coordinate permutations, and sign changes. In this
paper we study the $\mathcal{G}$-orbit structure of points on TIK3 and MK3
surfaces. Over finite fields, we study fibral connectivity and the existence of
large orbits, analogous to work of Bourgain, Gamburd, Sarnak and others for the
classical Markoff equation. For a particular $1$-parameter family of MK3
surfaces $\mathcal{W}_k$, we compute the full $\mathcal{G}$-orbit structure of
$\mathcal{W}_k(\mathbb{F}_p)$ for all primes $p\le113$, and we use this data as
a guide to find many finite $\mathcal{G}$-orbits in
$\mathcal{W}_k(\mathbb{C})$, including a family of orbits of size $288$
parameterized by a curve of genus $9$.
