  Recent experiments on $p$-orbital atomic bosons have suggested the emergence
of a spectacular ultracold superfluid with staggered orbital currents in
optical lattices. This raises fundamental questions like the effects of
collective thermal fluctuations, and how to directly observe such chiral order.
Here, we show via Monte Carlo simulations that thermal fluctuations destroy
this superfluid in an unexpected two-step process, unveiling an intermediate
normal phase with spontaneously broken time-reversal symmetry, dubbed "chiral
Bose liquid". For integer fillings ($n\geq 2$) in the chiral Mott regime,
thermal fluctuations are captured by an effective orbital Ising model, and
Onsager's powerful exact solution is adopted to determine the transition from
this intermediate liquid to the para-orbital normal phase at high temperature.
A suitable lattice quench is designed to convert the staggered angular
momentum, previously thought by experts difficult to directly probe, into
coherent orbital oscillations, providing a smoking-gun signature of chiral
order.
