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$\pi$ Men hosts a transiting super Earth ($P\approx6.27$ d, $m\approx4.82$ $M_{\oplus}$, $R\approx2.04$ $R_{\oplus}$) discovered by TESS and a cold Jupiter ($P\approx2093$ d, $m \sin I\approx10.02$ $M_{\rm{Jup}}$, $e\approx0.64$) discovered from radial velocity. We use Gaia DR2 and Hipparcos astrometry to derive the star's velocity caused by the orbiting planets and constrain the cold Jupiter's sky-projected inclination ($I_b=41-65^{\circ}$). From this we derive the mutual inclination ($\Delta I$) between the two planets, and find that $49^{\circ}< \Delta I < 131^{\circ}$ (1$\sigma$), and $28^{\circ} < \Delta I < 152^{\circ}$ (2$\sigma$). We examine the dynamics of the system using $N$-body simulations, and find that potentially large oscillations in the super Earth's eccentricity and inclination are suppressed by general relativistic precession. However, nodal precession of the inner orbit around the invariable plane causes the super Earth to only transit between 7-22 per cent of the time, and to usually be observed as misaligned with the stellar spin axis. We repeat our analysis for HAT-P-11, finding a large $\Delta I$ between its close-in Neptune and cold Jupiter and similar dynamics. $\pi$ Men and HAT-P-11 are prime examples of systems where dynamically hot outer planets excite their inner planets, with the effects of increasing planet eccentricities, planet-star misalignments, and potentially reducing the transit multiplicity. Formation of such systems likely involves scattering between multiple giant planets or misaligned protoplanetary discs. Future imaging of the faint debris disc in $\pi$ Men and precise constraints on the stellar spin orientation would provide strong tests for these formation scenarios., Comment: Corrected Fig 7 and Sec 6.1.1 (HAT-P-11 also weak planet-star coupling). Published as MNRAS 497, 2096-2118 (2020)