  Welschinger invariants enumerate real nodal rational curves in the plane or
in another real rational surface. We analyze the existence of similar
enumerative invariants that count real rational plane curves having prescribed
non-nodal singularities and passing through a generic conjugation-invariant
configuration of appropriately many points in the plane. We show that an
invariant like this is unique: it enumerates real rational three-cuspidal
quartics that pass through generically chosen four pairs of complex conjugate
points. Consequently, we show that through any generic configuration of four
pairs of complex conjugate points, one can always trace a pair of real rational
three-cuspidal quartics.
