  We present a radial velocity (RV) analysis of TOI-1136, a bright TESS system
with six confirmed transiting planets, and a seventh single-transiting planet
candidate. All planets in the system are amenable to transmission spectroscopy,
making TOI-1136 one of the best targets for intra-system comparison of
exoplanet atmospheres. TOI-1136 is young ($\sim$ 700 Myr), and the system
exhibits transit timing variations (TTVs). The youth of the system contributes
to high stellar variability on the order of 50 m s$^{-1}$, much larger than the
likely RV amplitude of any of the transiting exoplanets. Utilizing 359 HIRES
and APF RVs collected as a part of the TESS-Keck Survey (TKS), and 51 HARPS-N
RVs, we experiment with a joint TTV-RV fit. With seven possible transiting
planets, TTVs, more than 400 RVs, and a stellar activity model, we posit that
we may be presenting the most complex mass recovery of an exoplanet system in
the literature to date. By combining TTVs and RVs, we minimized GP overfitting
and retrieved new masses for this system: (m$_{b-g}$ = 3.50$^{+0.8}_{-0.7}$,
6.32$^{+1.1}_{-1.3}$, 8.35$^{+1.8}_{-1.6}$, 6.07$^{+1.09}_{-1.01}$,
9.7$^{+3.9}_{-3.7}$, 5.6$^{+4.1}_{-3.2}$ M$_{\oplus}$). We are unable to
significantly detect the mass of the seventh planet candidate in the RVs, but
we are able to loosely constrain a possible orbital period near 80 days. Future
TESS observations might confirm the existence of a seventh planet in the
system, better constrain the masses and orbital properties of the known
exoplanets, and generally shine light on this scientifically interesting
system.
