"Classical" partial deduction, within the framework by Lloyd and Shepherdson, computes partial deduction for separate atoms independently. As a consequence, a number of program optimisations, known from unfold/fold transformations and supercompilation, cannot be achieved. In this paper, we show that this restriction can be lifted through (polygenetic) specialisation of entire atom conjunctions. We present a generic algorithm for such partial deduction and discuss its correctness in an extended formal framework. We concentrate on novel control challenges specific to this "conjunctive" partial deduction. We refine the generic algorithm into a fully automatic concrete one that registers partially deduced conjunctions in a global tree, and prove its termination and correctness. We discuss some further control refinements and illustrate the operation of the concrete algorithm and/or some of its possible variants on interesting transformation examples.