This paper studies the effective behavior of piezoelectric and piezomagnetic circular fibrous composites with imperfect interfaces under longitudinal shear with in-plane electromagnetic fields. Two kinds of imperfect contact are investigated: mechanically stiff and dielectrically/magnetically highly conducting interfaces, and mechanically compliant and dielectrically/magnetically weakly conducting interfaces. For the former case, the potential field is continuous, while the normal component of the flux undergoes a discontinuity across the interface. For the latter case, the normal component of the flux is continuous, while there is a jump of potential field at such a contact. The classic work of Rayleigh (1892 Phil. Mag. 34 481-502) in a periodic conductive perfect composite is generalized to the current coupled magnetoelectroelastic composites with imperfect interfaces. It is shown that the expression of the effective property has exactly the same form as that in the ideal coupling composite. Finally, this method is used to study BaTiO 3-CoFe2O4 composites and provide insights into enhancing the effective magnetoelectric voltage coefficient by properly choosing the interface.