Precise phase control of microwaves and millimeter waves is critical for today's wireless communication and signal processing electronics. For decades, achieving even modest phase-shifting resolution has demanded a complex mix of transistor switches and passive electromagnetic structures. These true-phase delay, true-time-delay or quasi-true-time-delay circuits operate near resonances that heavily attenuate and distort signals, limiting transmission bandwidth. Despite numerous attempts, passive phase shifters have remained lossy and incompatible with high-channel-capacity beamforming, irrespective of the semiconductor fabrication process. In this article, we introduce a mechanism whereby waveguide reflectors based on coupled resonances can be reprogrammed to evade loss. Central to their operation is that loss from internal resonances is confined to low frequencies, while at high frequencies, their reflectivity is maximized and broadband phase variations, induced by those resonances, still persist. Since performance in the low-loss post-resonance spectrum is largely agnostic to the number of switches, ultra-fine digital tuning is possible. This breaks the historical tradeoff between loss and precision to achieve resolution surpassing state-of-the-art integrated circuits by over three orders (bits) of magnitude. The device does not distort the transmitted signal and consumes no power. Moreover, the chip occupies a sub-wavelength footprint in a Complementary Metal Oxide Semiconductor platform. This makes it an optimal candidate for seamless integration in on-chip multi-gigabit data links, radio astronomy transceivers and control hardware for millimeter-wave qubits.