Generic Vop\v{e}nka cardinals and models of ZF with few ${\mathrm{\aleph }}_1$-Suslin sets

We define a generic Vop\v{e}nka cardinal to be an inaccessible cardinal $\kappa$ such that for every first-order language $\mathcal{L}$ of cardinality less than $\kappa$ and every set $\mathcal{B}$ of $\mathcal{L}$-structures, if $\mathrm{\mid }\mathcal{B}\mathrm{\mid }=\kappa$ and every structure in $\mathcal{B}$ has cardinality less than $\kappa$, then an elementary embedding between two structures in $\mathcal{B}$ exists in some generic extension of $V$. We investigate connections between generic Vop\v{e}nka cardinals in models of ZFC and the number and complexity of ${\mathrm{\aleph }}_1$-Suslin sets of reals in models of ZF. In particular, we show that ZFC + (there is a generic Vop\v{e}nka cardinal) is equiconsistent with ZF + $(2^{{\mathrm{\aleph }}_1}\le ̸\mathrm{\mid }{S}_{{\mathrm{\aleph }}_1}\mathrm{\mid })$ where $S_{{\mathrm{\aleph }}_1}$ is the pointclass of all ${\mathrm{\aleph }}_1$-Suslin sets of reals, and also with ZF + $(S_{{\mathrm{\aleph }}_1}={\mathbf{\Sigma }}_2^1)$ + $(\mathrm{\Theta }={\mathrm{\aleph }}_2)$ where $\mathrm{\Theta }$ is the least ordinal that is not a surjective image of the reals.