  We construct infinite new classes of $AdS_4\times S^1\times S^5$ solutions of
type IIB string theory which have non-trivial $SL(2,\mathbb{Z})$ monodromy
along the $S^1$ direction. The solutions are supersymmetric and holographically
dual, generically, to $\mathcal{N}=1$ SCFTs in $d=3$. The solutions are first
constructed as $AdS_4\times \mathbb{R}$ solutions in $D=5$ $SO(6)$ gauged
supergravity and then uplifted to $D=10$. Unlike the known $AdS_4\times
\mathbb{R}$ S-fold solutions, there is no continuous symmetry associated with
the $\mathbb{R}$ direction. The solutions all arise as limiting cases of Janus
solutions of $d=4$, $\mathcal{N}=4$ SYM theory which are supported both by a
different value of the coupling constant on either side of the interface, as
well as by fermion and boson mass deformations. As special cases, the
construction recovers three known S-fold constructions, preserving
$\mathcal{N}=1,2$ and 4 supersymmetry, as well as a recently constructed
$\mathcal{N}=1$ $AdS_4\times S^1\times S^5$ solution (not S-folded). We also
present some novel "one-sided Janus" solutions that are non-singular.
