  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 $D=10$. 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, which
preserve $\mathcal{N}=1,2$ 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.
