 Solution

Let $A=\{a_1,\dots ,a_n\}\subseteq M$ and let $p(x/A)$ be a type. In an elementary extension choose $c$ realizing it and consider the empty-set type $q=\mathrm{tp}(a_1,\dots ,a_n,c)$. By hypothesis, $M$ contains a tuple $(b_1,\dots ,b_n,d)$ realizing $q$. The map $b_i\mapsto a_i$ is partial elementary, so omega-homogeneity extends it to include $d\mapsto d^{\prime }$. Then $d^{\prime }$ realizes $p$. Hence $M$ is omega-saturated, as stated by homogeneity plus realization of empty-set types implies saturation.