Towed ocean microstructure observations reveal that the temperature Cox number depends inversely on interface temperature-gradient Tz in the fingering-favorable thermohaline staircase east of Barbados. This implies fluxes independent of interface gradients. These conclusions are contrary to theoretical expectations from a finger Froude number stability criterion, |Δw|/N < 1.0: (i) Cox numbers independent of temperature gradient, and (ii) fluxes varying linearly with temperature gradient. We propose a hybrid wave/finger Froude number stability criterion, |(uzwx)(wyvz)|N2 ≤ 1.0 which reduces to Uz|Vw|/N2 ≤ 1.0 for salt sheets (δ/δx = 0, | Δw| = wy) aligned with background (internal-wave) shear Uz. The sheet criterion reproduces the observed behavior. For weak but nonzero background shears, 0.1 < Uz/N < 1.0, this constraint implies higher fluxes than unsheared square-planform fingers constrained by a finger Froude number | wxwy|/N2 < 1.0.