  The large spin-orbit misalignments in the DI Herculis stellar binary system
have resolved the decades-long puzzle of the anomalously slow apsidal
precession rate, but raise new questions regarding the origin of the
obliquities. This paper investigates obliquity evolution in stellar binaries
hosting modestly-inclined circumbinary disks. As the disk and binary axes
undergo mutual precession, each oblate star experiences a torque from its
companion star, so that the spin and orbital axes undergo mutual precession. As
the disk loses mass through a combination of winds and accretion, the system
may be captured into a high-obliquity Cassini state (a spin-orbit resonance).
The final obliquity depends on the details of the disk dispersal. We construct
a simple disk model to emulate disk dispersal due to viscous accretion and
photoevaporation, and identify the necessary disk properties for producing the
observed obliquities in DI Herculis. The disk must be massive (at least $10 \%$
of the binary mass). If accretion onto the binary is suppressed, the observed
high stellar obliquities are reproduced with a binary-disk inclination of $\sim
5^\circ - 10^\circ$, but if substantial accretion occurs, the inclination must
be larger, $\sim 20^\circ - 30^\circ$. If moderate accretion occurs, initially
the disk must lose its mass slowly, but eventually lose its remaining mass
abruptly, analogous to the observed two-timescale behavior for disks around
T-Tauri stars. The spin feedback on the binary orbit causes the binary-disk
inclination to decay as the obliquity evolves, a feature that is absent from
the standard Cassini state treatment.
