  Various studies have implied the existence of a gaseous halo around the
Galaxy extending out to 100 kpc. Galactic cosmic rays (CRs) that propagate to
the halo, either by diffusion or by convection with the possibly existing
large-scale Galactic wind, can interact with the gas therein and produce
gamma-rays via proton-proton collision. We calculate the cosmic ray
distribution in the halo and the gamma-ray flux, and explore the dependence of
the result on model parameters such as diffusion coefficient, CR luminosity, CR
spectral index. We find that the current measurement of isotropic gamma-ray
background at $\lesssim$TeV with Fermi Large Area Telescope already approaches
a level that can provide interesting constraints on the properties of Galactic
cosmic ray (e.g., with CR luminosity $L_{CR}\leq 10^{41}$erg/s). We also
discuss the possibilities of the Fermi bubble and IceCube neutrinos originating
from the proton-proton collision between cosmic rays and gas in the halo, as
well as the implication of our results for the baryon budget of the hot
circumgalactic medium of our Galaxy. Given that the isotropic gamma-ray
background is likely to be dominated by unresolved extragalactic sources,
future telescopes may extract more individual sources from the IGRB, and hence
put even more stringent restriction on the relevant quantities (such as
Galactic cosmic ray luminosity and baryon budget in the halo) in the presence
of a turbulent halo that we consider.
