  We investigate supersymmetric flows in type IIB supergravity that preserve an
$SO(2,2)$ space-time symmetry and asymptote to $AdS_{5} \times S^5$ at both
endpoints. The flows are constructed as Janus-type $\mathbb{R} \times AdS_{3}$
BPS domain-walls in the effective four-dimensional
$[SO(1,1)\times{S}O(6)]\ltimes\mathbb{R}^{12}$ gauged maximal supergravity
describing the massless sector of type IIB supergravity compactified on $S^1
\times S^{5}$. The compactification includes an S-duality hyperbolic twist
along the $S^1$ which, when combined with an appropriate choice of boundary
conditions for the running scalars, generates special flows that develop an
S-fold regime at their core, thus enhancing the space-time symmetry there to
$SO(2,3)$. Via the AdS/CFT correspondence, the flows constructed here are
conjectured to describe conformal interfaces in a circle compactification of
$\mathcal{N}=4$ $\textrm{SYM}_{4}$.
