  The $SU(2)_L\times U(1)_Y$ invariance of the Standard Model Effective Field
Theory (SMEFT) predicts multiple restrictions in the space of Wilson
coefficients of $U(1)_{em}$ invariant effective lagrangians such as the
Low-energy Effective Field Theory (LEFT), used for low-energy flavor-physics
observables, or the Higgs Effective Field Theory (HEFT) in unitary gauge,
appropriate for weak-scale observables. In this work, we derive and list all
such predictions for semileptonic operators up to dimension 6. We find that
these predictions can be expressed as 2223 linear relations among the HEFT/LEFT
Wilson coefficients, that are completely independent of any assumptions about
the alignment of the mass and flavor bases. These relations connect diverse
experimental searches such as rare meson decays, high-$p_T$ dilepton searches,
top decays, $Z$-pole observables, charged lepton flavor violating observables
and non-standard neutrino interaction searches. We demonstrate how these
relations can be used to derive strong indirect constraints on multiple Wilson
coefficients that are currently either weakly constrained from direct
experiments or have no direct bound at all. These relations also imply, in
general, that evidence for new physics in a particular search channel must be
accompanied by correlated anomalies in other channels.
