  Given an essential semilattice congruence $\equiv$ on the left weak order of
a Coxeter group $W$, we define the Coxeter stack-sorting operator ${\bf
S}_\equiv:W\to W$ by ${\bf
S}_\equiv(w)=w\left(\pi_\downarrow^\equiv(w)\right)^{-1}$, where
$\pi_\downarrow^\equiv(w)$ is the unique minimal element of the congruence
class of $\equiv$ containing $w$. When $\equiv$ is the sylvester congruence on
the symmetric group $S_n$, the operator ${\bf S}_\equiv$ is West's
stack-sorting map. When $\equiv$ is the descent congruence on $S_n$, the
operator ${\bf S}_\equiv$ is the pop-stack-sorting map. We establish several
general results about Coxeter stack-sorting operators, especially those acting
on symmetric groups. For example, we prove that if $\equiv$ is an essential
lattice congruence on $S_n$, then every permutation in the image of ${\bf
S}_\equiv$ has at most $\left\lfloor\frac{2(n-1)}{3}\right\rfloor$ right
descents; we also show that this bound is tight.
  We then introduce analogues of permutree congruences in types $B$ and
$\widetilde A$ and use them to isolate Coxeter stack-sorting operators
$\mathtt{s}_B$ and $\widetilde{\hspace{.05cm}\mathtt{s}}$ that serve as
canonical type-$B$ and type-$\widetilde A$ counterparts of West's stack-sorting
map. We prove analogues of many known results about West's stack-sorting map
for the new operators $\mathtt{s}_B$ and
$\widetilde{\hspace{.05cm}\mathtt{s}}$. For example, in type $\widetilde A$, we
obtain an analogue of Zeilberger's classical formula for the number of
$2$-stack-sortable permutations in $S_n$.
