For a stationary wave,

The equation of a stationary wave can be written as $y = A \sin(kx) \cos(\omega t)$.

The term $A \sin(kx)$ gives the amplitude of oscillation for a particle at position $x$.

Between two consecutive nodes, the value of $kx$ lies in the interval $(n\pi, (n+1)\pi)$. In this region, the sign of $\sin(kx)$ remains the same. Hence, all particles located between two consecutive nodes oscillate in phase.

When crossing a node, the sign of $\sin(kx)$ changes, which means particles in adjacent loops oscillate out of phase by $\pi$.

Since the sign of $\sin(kx)$ alternates for consecutive loops, particles in alternate loops (and thus alternate antinodes) have the same sign and oscillate in phase.