The expansion of offshore wind energy results in more wake influences between wind farms and wind farm clusters, which should be mitigated to reduce negative impacts on the power production. In this study we assess whether vertical staggering of wind turbine heights within a wind farm could mitigate farm wake interaction between two neighbouring wind farms. The aim is to understand how wake recovery mechanisms and three-dimensional wake characteristics differ between vertically staggered and non-staggered wind farms. We performed large-eddy simulations of a pair of an upstream and a downstream wind farm in a conventionally neutral atmospheric boundary layer and compared several vertical staggering layouts at the upstream wind farm to the non-staggered reference. To assess differences, we analysed mean wake characteristics, the three-dimensional flow field, the wake recovery budget, and power production. We found that vertical staggering hindered the mean wake recovery of the horizontal wind speed at the hub height of the lower turbines by reducing momentum at higher altitudes. Investigating spatial wind fields revealed a more heterogeneous, narrower farm wake with streaks of increased wind speeds for vertically staggered wind farms. The wake differences were explained by the formation of larger-scale advective flow structures for vertically staggering cases and their larger contribution to the mean wake recovery compared to turbulent mixing. A circulation zone at the northern wake edge was present for staggered and non-staggered setups, which caused a streak of increased wind speeds in the lower half of the boundary layer. The wake heterogeneities persisted far downstream and impacted the wake and power production of the non-staggered, downstream wind farm, causing locally increased or decreased power. The total power production of the vertically staggered upstream wind farm was strongly increased due to higher wind speeds at the taller turbines, a lower rotor density per height, and a better wake recovery at lower hub heights inside the wind farm To conclude, we showed that vertical staggering has the potential of locally improving the farm wake recovery by creating or enhancing large-scale advective flow structures, influencing the wake and power of a downstream wind farm. The advection-driven recovery of the far farm wake is relevant for wind farms beyond vertically staggered setups, requiring future analyses to focus on production mechanisms of large-scale advective structures.