  We define an integer-valued virtual count of embedded pseudo-holomorphic
curves of two times a primitive homology class and arbitrary genus in
symplectic Calabi--Yau $3$-folds, which can be viewed as an extension of
Taubes' Gromov invariant. The construction depends on a detailed study of
bifurcations of moduli spaces of embedded pseudo-holomorphic curves which is
partially motivated by Wendl's recent solution of Bryan--Pandharipande's
super-rigidity conjecture.
