${\text{NS}}_{{\omega }_1}$ saturated, ${\mathrm{\Delta }}_1(\{{\omega }_1\})$-definable and a ${\mathrm{\Delta }}_4^1$-definable well-order of the reals

Assuming $M_1$, the canonical inner model with one Woodin cardinal exists, we construct a model in which the nonstationary ideal on ${\omega }_1$ is ${\mathrm{\aleph }}_2$-saturated, ${\mathrm{\Delta }}_1$-definable with ${\omega }_1$ as the only parameter and there is a ${\mathrm{\Sigma }}_4^1$-definable well-order of the reals. This implies that contrary to the assumption that $N{S}_{{\omega }_1}$ is ${\mathrm{\aleph }}_1$-dense, the assumption of $N{S}_{{\omega }_1}$ being saturated and ${\mathrm{\Delta }}_1$-definable does not imply any nice structural properties for the projective subsets of the reals.