  We study the three-dimensional U(1) lattice gauge theory of a CP$^1$ spinon
(Schwinger boson) field and a Higgs field. It is a bosonic $t$-$J$ model in
slave-particle representation, describing the antiferromagnetic (AF) Heisenberg
spin model with doped bosonic holes expressed by the Higgs field. The spinon
coupling term of the action favors AF long-range order, whereas the holon
hopping term in the ferromagnetic channel favors Bose-Einstein condensation
(BEC) of holons. We investigate the phase structure by means of Monte-Carlo
simulations and study an interplay of AF order and BEC of holes. We consider
the two variations of the model; (i) the three-dimensional model at finite
temperatures, and (ii) the two-dimensional model at vanishing temperature. In
the model (i) we find that the AF order and BEC coexist at low temperatures and
certain hole concentrations. In the model (ii), by varying the hole
concentration and the stiffness of AF spin coupling, we find a phase diagram
similar to the model (i). Implications of the results to systems of cold atoms
and the fermionic $t$-$J$ model of strongly-correlated electrons are discussed.
