As far as I can tell Newton's third law says "actio = reactio" which refers to the mutual forces between the two particles, not to the acceleration directly. But from this it follows that the ratio of acceleration of the two particles must the constant without external forces acting on the particles because: `F1 = m1 * a1` and `F2 = m2 * a2`. Due to actio = reactio: `F1 = -F2`. Hence: `a1/a2 = -m2/m1` which is constant if the particle masses are constant (equations are meant as one per spatial direction).

Honestly, I've never heard of a "consistency relation" in classical mechanics, so I can't comment the second part of your question.