uncertainty principle dictates that incompatible observables, such as position and momentum, cannot both take on arbitrarily precise values. But how does a wavefunction evolve when two such quantities are probed simultaneously, and how does the uncertainty principle dynamically inhibit precise measurement outcomes? To realize this unexplored regime, we simultaneously apply two continuous quantum non-demolition probes of non-commuting observables on a superconducting qubit. We achieve this capability by developing a novel measurement scheme that allows us to control the axes of multiple readout channels. We show that the uncertainty principle directly governs the dynamics of the state, and consequently standard wavefunction collapse is replaced by a persistent diffusion that exhibits several distinct regimes. Although evolution of the state now differs drastically from that of a conventional measurement, information about both non-commuting observables is extracted by keeping track of the time ordering of the measurement record, enabling quantum state tomography without alternating measurements. Our work creates new capabilities for quantum control, including rapid state purification, adaptive measurement, measurement-based state steering and continuous quantum error correction. As physical quantum systems interact with their environments via non-commuting degrees of freedom, our work offers a new, more natural approach to experimentally study contemporary quantum foundations.