Magnetic Microphase Inhomogeneity as a Thermodynamic Precursor of Ground State Phase Separation in Weakly Coupled Spin-$\frac{3}{2}$ Chains

$\gamma$-CoV$_{2}$O$_{6}$ is a quasi one-dimensional spin-$\frac{3}{2}$ magnet that possesses two distinct magnetic orders in the ground state with modulation vectors $k_\mathrm{1}$ = ($\frac{1}{2}$, 0, 0) and $k_\mathrm{2}$ = ($\frac{1}{4}$, 0, -$\frac{1}{4}$), respectively. Here, we use muon spin relaxation and rotation to reveal the thermodynamics of the magnetic phase separation in this compound. In the paramagnetic (PM) region, short-range correlated spin clusters emerge at $T_\mathrm{m}$ $\simeq$ 26 K at the $\it{partial}$ expense of the PM volume. Upon further cooling, we show that these emergent clusters become spatially coherent at $T_\mathrm{{N2}}$ = 7.5 K and eventually form the $k_\mathrm{2}$ order at $T^{\star}$ = 5.6 K, while the remaining PM spins are driven into the $k_\mathrm{1}$ state at $T_\mathrm{{N1}}$ = 6.6 K. These results stress magnetic microphase inhomogeneity as a thermodynamic precursor for the ground state phase separation in weakly coupled spin-$\frac{3}{2}$ chains.