  We describe a new approach to precision NMR with hyperpolarized gases
designed to mitigate NMR frequency shifts due to the alkali spin exchange
field. The electronic spin polarization of optically pumped alkali atoms is
square-wave modulated at the noble-gas NMR frequency and oriented transverse to
the DC Fourier component of the NMR bias field. Noble gas NMR is driven by
spin-exchange collisions with the oscillating electron spins. On resonance, the
time-average torque from the oscillating spin-exchange field produced by the
alkali spins is zero. Implementing the NMR bias field as a sequence of alkali
2$ \pi $-pulses enables synchronization of the alkali and noble gas spins
despite a 1000-fold discrepancy in gyromagnetic ratio. We demonstrate this
method with Rb and Xe, and observe novel NMR broadening effects due to the
transverse oscillating spin exchange field. When uncompensated, the
spin-exchange field at high density broadens the NMR linewidth by an order of
magnitude, with an even more dramatic suppression (up to 70x) of the phase
shift between the precessing alkali and Xe polarizations. When we introduce a
transverse compensation field, we are able to eliminate the spin-exchange
broadening and restore the usual NMR phase sensitivity. The projected
quantum-limited sensitivity is better than 1 nHz/$\sqrt{\rm Hz}$.
