We address in this paper the issue of leadership when two governments provide public goods to their constituencies with cross-border externalities as both public goods are valued by consumers in both countries. We study a timing game between two different countries: before providing public goods, the two policymakers non-cooperatively decide their preferred sequence of moves. We establish conditions under which a first- or second-mover advantage emerges for each country, highlighting the role of spillovers and the complementarity or substitutability of public goods. As a result, we are able to prove that there is no leader when, for both countries, public goods are substitutable. When public goods are complements for both countries, each of them may emerge as the leader in the game. Hence a coordination issue arises. We use the notion of risk-dominance to select the leading government. Finally, in the mixed case, the government for whom public goods are substitutable becomes the leader.