We can model disjunctions (note I am not looking for convex hull) of $t$ unbounded convex polyhedra given by $A^{(1)}x^{(1)}\leq b^{(1)}$,$\dots$,$A^{(t)}x^{(t)}\leq b^{(t)}$ exactly with a mixed integer linear or convex program with $O(t)$ integer variables and perhaps $2^{O(t)}$ real variables?

Let the $i$th convex polyhedra be $A^{(i)}x^{(i)}\leq b^{(i)}$ where $A^{(i)}\in\mathbb R^{m_i\times n}$ and $b^{(i)}\in\mathbb R^{m_i}$ are fixed. Then if we introduce binary variables $y_1,\dots,y_t\in\{0,1\}$ and a real vector $x=\sum_{i=1}^tx^{(i)}\in\mathbb R^n$ then
$$A^{(i)}x^{(i)}\leq b^{(i)}y_i$$
$$y_1+\dots+y_t=1$$
$$x=\sum_{i=1}^tx^{(i)}\in\mathbb R^n$$
suffices.

However the trick breaks down if $b^{(i)}=0$ at $i\in\{1,\dots,t\}$. That is if $b^{(i)}$ are $0$ vectors then the trick breaks down.

If the $t$ polytopes are given by $A^{(i)}x^{(i)}\leq0$ where $A^{(i)}\in\mathbb R^{m_i\times n}$ and $b^{(i)}\in\mathbb R^{m_i}$ are fixed then how do we model unions? Is there a standard trick with at least convex convex constraints introduced and at least when $x^{(i)}\in[0,1]^n$ holds?