  When two layers of two-dimensional materials are assembled with a relative
twist, moir\'e patterns arise, inducing a tremendous wealth of exotic
phenomena. In this work, we consider twisting two triangular lattices hosting
Dirac quantum spin liquids. A single decoupled layer is described by compact
quantum electrodynamics in 2+1 dimensions (QED$_3$) with an emergent
$\mathrm{U}(1)$ gauge field, which is assumed to flow to a strongly interacting
fixed point in the IR with conformal symmetry. We use recent results for the
quantum numbers of monopole operators, which tunnel $2 \pi$ fluxes of the
compact gauge field. It is found that, in the bilayer system, interlayer
monopole tunneling is a symmetry-allowed relevant perturbation which induces an
(ordering) instability. We show using perturbation theory that upon twisting
the two layers the system remains unstable under the interlayer interaction,
but any finite twist angle softens this instability compared to the untwisted
case. To analyse the resulting phase induced by the (twisted) interlayer
tunneling, we use "conformal mean field theory", which reduces the interacting
bilayer system to two copies of QED$_3$ coupled to background fields which are
to be determined self-consistently. In the weak-coupling regime, where the
interlayer coupling is weak compared to the energy scale set by the moir\'e
lattice constant, we solve the self-consistency equations perturbatively. In
the limit of strongly coupled layers, a local scaling approximation is used,
and we find that the magnetically ordered state exhibits a lattice of magnetic
vortices, with the lattice constant tunable through the twisting angle.
