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.

As far as I can tell, Newton's third law says "action = reaction" 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: $F_1 = m_1 \cdot a_1$ and $F_2 = m_2 \cdot a_2$. Due to action = reaction: $F_1 = -F_2$. Hence: $a_1/a_2 = -m_2/m_1$, 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.