  Ultracold atoms loaded into higher Bloch bands provide an elegant setting for
realizing many-body quantum states that spontaneously break time-reversal
symmetry through the formation of chiral orbital order. The applicability of
this strategy remains nonetheless limited due to the finite lifetime of atoms
in high-energy bands. Here we introduce an alternative framework, suitable for
bosonic gases, which builds on assembling square plaquettes pierced by a
$\pi$-flux (half a magnetic-flux quantum). This setting is shown to be formally
equivalent to an interacting bosonic gas loaded into $p$ orbitals, and we
explore the consequences of the resulting chiral orbital order, both for weak
and strong onsite interactions. We demonstrate the emergence of a chiral
superfluid vortex lattice, exhibiting a long-lived gapped collective mode that
is characterized by local chiral currents. This chiral superfluid phase is
shown to undergo a phase transition to a chiral Mott insulator for sufficiently
strong interactions. Our work establishes coupled $\pi$-flux plaquettes as a
practical route for the emergence of orbital order and chiral phases of matter.
