  We say that an element $g$ of a group $G$ is almost right Engel if there is a
finite set ${\mathscr R}(g)$ such that for every $x\in G$ all sufficiently long
commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$, that is, for
every $x\in G$ there is a positive integer $n(x,g)$ such that
$[...[[g,x],x],\dots ,x]\in {\mathscr R}(g)$ if $x$ is repeated at least
$n(x,g)$ times. Thus, $g$ is a right Engel element precisely when we can choose
${\mathscr R}(g)=\{ 1\}$.
  We prove that if all elements of a compact (Hausdorff) group $G$ are almost
right Engel, then $G$ has a finite normal subgroup $N$ such that $G/N$ is
locally nilpotent. If in addition there is a uniform bound $|{\mathscr
R}(g)|\leq m$ for the orders of the corresponding sets, then the subgroup $N$
can be chosen of order bounded in terms of $m$. The proofs use the
Wilson--Zelmanov theorem saying that Engel profinite groups are locally
nilpotent and previous results of the authors about compact groups all elements
of which are almost left Engel.
