The strong geodetic problem is a recent variation of the geodetic problem. For a graph G, its strong geodetic number sg(G) is the cardinality of a smallest vertex subset S, such that each vertex of G lies on a fixed shortest path between a pair of vertices from S. In this paper, the strong geodetic problem is studied on the Cartesian product of graphs. A general upper bound for sg(G □ H) is determined, as well as exact values for Km □ Kn, K1,k □ Pl, and prisms over Kn–e. Connections between the strong geodetic number of a graph and its subgraphs are also discussed.