  The combination of optical tweezer arrays with strong interactions -- via
dipole-exchange of molecules and van-der-Waals interactions of Rydberg atoms --
has opened the door for the exploration of a wide variety of quantum spin
models. A next significant step will be the combination of such settings with
mobile dopants: This will enable to simulate the physics believed to underlie
many strongly correlated quantum materials. Here we propose an experimental
scheme to realize bosonic t-J models via encoding the local Hilbert space in a
set of three internal atomic or molecular states. By engineering
antiferromagnetic (AFM) couplings between spins, competition between charge
motion and magnetic order similar to that in high-$T_c$ cuprates can be
realized. Since the ground states of the 2D bosonic AFM t-J model we propose to
realize have not been studied extensively before, we start by analyzing the
case of two dopants -- the simplest instance in which their bosonic statistics
plays a role, and contrast our results to the fermionic case. We perform
large-scale density matrix renormalization group (DMRG) calculations on
six-legged cylinders, and find a strong tendency for bosonic holes to form
stripes. This demonstrates that bosonic, AFM t-J models may contain similar
physics as the collective phases in strongly correlated electrons.
