  We study the conditions under which 4d $\mathcal{N}=2$ superconformal field
theories (SCFTs) have multiplets housing operators that are chiral with respect
to an $\mathcal{N}=1$ subalgebra. Our main focus is on the set of often-ignored
and relatively poorly understood $\overline{\mathcal{B}}$ representations.
These multiplets typically evade direct detection by the most popular
non-perturbative 4d $\mathcal{N}=2$ tools and correspondences. In spite of this
fact, we demonstrate the ubiquity of $\overline{\mathcal{B}}$ multiplets and
show they are associated with interesting phenomena. For example, we give a
purely algebraic proof that they are present in all local unitary
$\mathcal{N}>2$ SCFTs. We also show that $\overline{\mathcal{B}}$ multiplets
exist in $\mathcal{N}=2$ theories with rank greater than one and a conformal
manifold or a freely generated Coulomb branch. Using recent topological quantum
field theory results, we argue that certain $\overline{\mathcal{B}}$ multiplets
exist in broad classes of theories with the $\mathbb{Z}_2$-valued 't Hooft
anomaly for $Sp(N)$ global symmetry. Motivated by these statements, we then
study the question of whether $\overline{\mathcal{B}}$ multiplets exist in
rank-one SCFTs with exactly $\mathcal{N}=2$ SUSY. We conclude with various open
questions.
