Return to Question

1. BOTTOM UP. Start from a minimal model $W_0$ such that $G\notin W$ (for instance the constructibles in $M$) , and look at the set of extensions $W$ of the bottom $W_0$ such that $W[A] \neq M$, ordered by inclusion, then try to take the colimit of this ordered set (in other words, you hope that the sup of all of them is a model and does not contain A, but adding A you get M) . Is such a beast exists you found your A-eroded M
2. TOP DOWN See Jonas's answer (I would call it the "take the limit " method).

2 is actually more in line with erosion, getting rid of as much as you can, as so aptly Jonas said.

1. BOTTOM UP. Start from a minimal model $W_0$ such that $A\notin W$ (for instance the constructibles in $M$) , and look at the set of extensions $W$ of the bottom $W_0$ such that $W[A] \neq M$, ordered by inclusion, then try to take the colimit of this ordered set (in other words, you hope that the sup of all of them is a model and does not contain A, but adding A you get M) . Is such a beast exists you found your A-eroded M
2. TOP DOWN See Jonas's answer (I would call it the "take the limit " method).

2 is actually more in line with erosion, getting rid of as much as you can, as so aptly Jonas said.

So, either going from non A-grounds and looking for their union, or from A-grounds and looking for their intersection

1. BOTTOM UP. Start from a minimal model $W$ such that $G\notin W$ (for instance the constructibles in $M$) , and look at the set of any forcing extension of the bottom $W$ ordered by inclusion, then try to take the colimit of this ordered set (in other words, you hope that the sup of all of them is a model and does not contain A) . Is such a beast exists you found your A-eroded M
2. TOP DOWN See Jonas's answer (I would call it the "take the limit " method).

1. BOTTOM UP. Start from a minimal model $W_0$ such that $G\notin W$ (for instance the constructibles in $M$) , and look at the set of extensions $W$ of the bottom $W_0$ such that $W[A] \neq M$, ordered by inclusion, then try to take the colimit of this ordered set (in other words, you hope that the sup of all of them is a model and does not contain A, but adding A you get M) . Is such a beast exists you found your A-eroded M
2. TOP DOWN See Jonas's answer (I would call it the "take the limit " method).

1. BOTTOM UP. Start from a minimal model $W$ such that $G\notin W$ (for instance the constructibles in $M$ , and look at the directed set of any such $W$ ordered by inclusion, then try to take the colimit of this ordered set (in other words, you hope that the sup of all of them is a model and does not contain A) . Is such a beast exists you found your A-eroded M
2. TOP DOWN See Jonas's answer (I would call it the "take the limit " method

2 is actually more in line with erosion, getting rid of as much as you can.

1. BOTTOM UP. Start from a minimal model $W$ such that $G\notin W$ (for instance the constructibles in $M$) , and look at the set of any forcing extension of the bottom $W$ ordered by inclusion, then try to take the colimit of this ordered set (in other words, you hope that the sup of all of them is a model and does not contain A) . Is such a beast exists you found your A-eroded M
2. TOP DOWN See Jonas's answer (I would call it the "take the limit " method).

2 is actually more in line with erosion, getting rid of as much as you can, as so aptly Jonas said.