In this article, a duality approach to multiobjective H2/H problems is pursued in which real-rational, para-Hermitian multipliers and real-valued ones are associated to H and (as usual) H2 constraints, respectively. It is shown that the maximisation of a dual functional over all such multipliers yields the optimal value of the original multiobjective H2/H problem. To compute lower bounds on the latter and the corresponding approximate solutions to the original problem, the maximisation of the dual functional over linearly-parameterised, finite-dimensional classes of real-rational multipliers is shown to be equivalent to semi-definite, linear programming problems - once the optimal multipliers in such a class are obtained, the corresponding approximate solutions can be computed from an unconstrained H2 problem. Iterative modification of such classes is discussed to obtain increasing sequences of lower bounds on the optimal value of the original problem. This is done on the basis of (locally) increasing directions for the dual functional which go beyond the finite-dimensional class of multipliers considered in a given step. Finally, a numerical example is presented to illustrate the way the presented results can lead to approximate solutions to the multiobjective H2/H problem together with tight estimates of the corresponding deviation from its optimal value.