  HAT-P-11 is a mid-K dwarf that hosts one of the first Neptune-sized planets
found outside the solar system. The orbit of HAT-P-11b is misaligned with the
star's spin --- one of the few known cases of a misaligned planet orbiting a
star less massive than the Sun. We find an additional planet in the system
based on a decade of precision radial velocity (RV) measurements from
Keck/HIRES. HAT-P-11c is similar to Jupiter in its mass ($M_P \sin{i} =
1.6\pm0.1$ $M_J$) and orbital period ($P = 9.3^{+1.0}_{-0.5}$ year), but has a
much more eccentric orbit ($e=0.60\pm0.03$). In our joint modeling of RV and
stellar activity, we found an activity-induced RV signal of $\sim$7 m s$^{-1}$,
consistent with other active K dwarfs, but significantly smaller than the 31 m
s$^{-1}$ reflex motion due to HAT-P-11c. We investigated the dynamical coupling
between HAT-P-11b and c as a possible explanation for HAT-P-11b's misaligned
orbit, finding that planet-planet Kozai interactions cannot tilt planet b's
orbit due to general relativistic precession; however, nodal precession
operating on million year timescales is a viable mechanism to explain
HAT-P-11b's high obliquity. This leaves open the question of why HAT-P-11c may
have such a tilted orbit. At a distance of 38 pc, the HAT-P-11 system offers
rich opportunities for further exoplanet characterization through astrometry
and direct imaging.
