Experiments on the intrinsic stress in immiscible Cu/W nanomultilayers (NMLs) show two unusual features: a compressive-to-tensile transition as a function of NML deposition conditions and extremely high interface stresses under both high- and low-pressure deposition. Since intrinsic stresses in as-fabricated NMLs have critical consequences for component performance, these unusual results must be understood so that they can be controlled. Here, atomistic simulations with a chemically-accurate neural network potential reveal that large tensile stresses arise due to interfacial intermixing that occurs at high deposition pressures (pAr> 1 Pa) while large compressive stresses arise due to metastable bcc Cu-W alloy formation that occurs at low deposition pressures (pAr< 1 Pa). Experimental evidence for the metastable bcc phases further supports the simulation results. These findings have significant implications in all modern technologies with heterogeneous interfaces, especially where kinetically frozen metastable interface states associated with deposition conditions cannot be eliminated by high-temperature annealing.