The command Union(cs1, cs2, R) returns a constructible set, the union of cs1 and cs2.

There might be redundancy in the output.

This command is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form Union(..) only after executing the command with(RegularChains[ConstructibleSetTools]).  However, it can always be accessed through the long form of the command by using RegularChains[ConstructibleSetTools][Union](..).

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$

$\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$F≔2x^2+3xy+y^2-3x-3y$

$F≔2x^2+3xy+y^2-3x-3y$

$G≔x{y}^2-x$

$G≔x{y}^2-x$

$H≔y^3-y$

$H≔y^3-y$

$\mathrm{cs1}≔\mathrm{GeneralConstruct}\left(\left[F\,G\right]\,\left[1\right]\,R\right)$

$\mathrm{cs1}≔\mathrm{constructible\_set}$

$\mathrm{cs2}≔\mathrm{GeneralConstruct}\left(\left[G\,H\right]\,\left[1\right]\,R\right)$

$\mathrm{cs2}≔\mathrm{constructible\_set}$

$\mathrm{cs3}≔\mathrm{Union}\left(\mathrm{cs1}\,\mathrm{cs2}\,R\right)\:$$\mathrm{Info}\left(\mathrm{cs3}\,R\right)$

$\left[\left[x-2\,y+1\right]\,\left[1\right]\right],\left[\left[x-1\,y-1\right]\,\left[1\right]\right],\left[\left[x+1\,y-1\right]\,\left[1\right]\right],\left[\left[x\,y-3\right]\,\left[1\right]\right],\left[\left[x-1\,y+1\right]\,\left[1\right]\right],\left[\left[y-1\right]\,\left[1\right]\right],\left[\left[y+1\right]\,\left[1\right]\right],\left[\left[x\,y\right]\,\left[1\right]\right]$

$\mathrm{cs3}≔\mathrm{MakePairwiseDisjoint}\left(\mathrm{cs3}\,R\right)$

$\mathrm{cs3}≔\mathrm{constructible\_set}$

$\mathrm{Info}\left(\mathrm{cs3}\,R\right)$

$\left[\left[x\,y-3\right]\,\left[1\right]\right],\left[\left[x\,y\right]\,\left[1\right]\right],\left[\left[y+1\right]\,\left[1\right]\right],\left[\left[y-1\right]\,\left[1\right]\right]$