This paper presents the first three-dimensional (3D) particle-in-cell (PIC) simulations of castellated plasma-facing components (PFCs) in tokamaks. Special focus is given to crossings between poloidal and toroidal gaps where elevated heat loads are expected to occur. Moreover, the crossings may affect the plasma penetration into the gaps between tiles. Both of these problems are of high importance for ITER when estimating the lifetime of its PFCs. Localized heat loads can potentially lead to damage of the tiles, while the plasma penetration is related to fuel retention in the gaps due to redeposition of eroded wall material. This problem has previously been targeted by 2D PIC simulations using our in-house code SPICE2, where toroidal and poloidal gaps (PGs) had to be simulated separately. This paper presents the results of a full 3D3V code SPICE3, which allows us to simulate a more realistic geometry of the tiles including the gap crossings and includes the complete E × B drift, which could not be simulated in 2D. The results of self-consistent simulations show that the crossing acts as a transport channel for electrons, allowing them to enter the plasma shadowed region in PGs. As a consequence, the potential near the gap entrance is modified allowing more ions to flow deep inside the gap. The combination of the plasma flow and an E × B drift in the crossing directs ions onto one tile corner, which receives elevated heat load.