In a Cournot duopoly model in which exporters compete in a third market, this paper revisits the classical issue (dating back to the pioneering work of Brander and Spencer, Export Share and International Market Share Rivalry, 1985) of the strategic trade policy choice in the presence of the passive participation of one firm in the rival. Passive cross-ownership dramatically alters the participating and participated firms' governments' choice to apply the strategic trade policy instrument, the equilibria typology and their efficiency properties. In fact, if the share of cross-ownership is sufficiently large, the participated firm's government finds optimal to tax export. Moreover, beyond an adequately high threshold, cross-ownership modifies the equilibrium from the activist regime for both countries to an asymmetric regime in which only the participating firm's government intervenes. In addition, in the case of the traditional common activist regime equilibrium, the classical prisoner's dilemma game structure may disappear.