  We investigate the admissible vector-valued modular forms having three
independent characters and vanishing Wronskian index and determine which ones
correspond to genuine 2d conformal field theories. This is done by finding
bilinear coset-type relations that pair them into meromorphic characters with
central charges 8, 16, 24, 32 and 40. Such pairings allow us to identify some
characters with definite CFTs and rule out others. As a key result we classify
all unitary three-character CFT with vanishing Wronskian index, excluding
$c=8,16$. The complete list has two infinite affine series $B_{r,1},D_{r,1}$
and 45 additional theories. As a by-product, at higher values of the total
central charge we also find constraints on the existence or otherwise of
meromorphic theories. We separately list several cases that potentially
correspond to Intermediate Vertex Operator Algebras.
