Estimating forest aboveground biomass (AGB) from Earth observation combines two structurally incompatible label sources: spaceborne lidar provides canopy structure at millions of locations but no biomass estimate, and ground-based plots provide biomass at thousands of biased locations but no metrics of structure. No single training sample carries labels for all target variables, plot labels are missing not at random (MNAR), and biomass is linked to the structural variables by known but biome-specific allometric laws. We formalise this as multi-task dense regression under heterogeneous disjoint partial supervision with MNAR labels and inter-task physical constraints, and propose StruMPL to address it jointly. A shared encoder feeds per-variable regression, imputation, and propensity heads for spatial MNAR correction, and a learnable physics module that evaluates the inter-task constraint on the model's own predictions at every pixel. The supervised loss uses an Augmented IPW (AIPW) pseudo-outcome with stop-gradients on the propensity and on the imputation baseline; we show analytically and empirically that both are necessary for joint optimisation to recover IPW-weighted stationary points while keeping the loss bounded. On two ecologically distinct biomes, StruMPL outperforms ablation variants and the closest published method on AGB RMSE and bias, with a stratified analysis showing AIPW reduces high-AGB bias by ~54%.