  In the absence of parity and time-reversal symmetries, insulators can exhibit
magnetoelectric responses, in which applied magnetic fields induce charge
polarization and, conversely, applied electric fields induce magnetization.
While there is a long history of the study of magnetoelectric response in
fermionic insulators, the same for bosonic insulators has been limited. We
consider the magnetoelectric response in lattice insulators built out of
charged bosonic degrees of freedom and derive a bulk formula for the
corresponding linear response tensor. The resulting formulae feature several
contributions including a Chern-Simons integral over the bands of the bosonic
excitations. We construct several minimal microscopic models that illustrate
the ingredients required to obtain a sizable bosonic magnetoelectric response.
Our formalism can be applied to bosonic Mott insulators subject to synthetic
gauge fields and/or tilted potentials as well as to the spinon sector in the
Coulomb phase of a $U(1)$ quantum spin liquid.
