Can we apply the equivalence principle in electromagnetics to multiple regions simultaneously?

Setup 1

An object $$\(\epsilon, \mu)\$$ occupies volume $$\V_1\$$ and is enclosed by surface $$\S_1\$$. It sits in vacuum $$\(\epsilon_0, \mu_0)\$$, denoted by $$\V_0\$$. In $$\V_0\$$ there is an electric field $$\\vec{E}_\mathrm{inc}\$$ which impinges on said object. As a result, a set of fields $$\(\vec{E}_1, \vec{H}_1)\$$ is established inside $$\V_1\$$.

We know that we can introduce equivalent current sources on $$\S_1\$$, which generate the correct fields $$\(\vec{E}_1, \vec{H}_1)\$$ inside $$\V_1\$$, but exactly cancel ("extinct") the field outside, so that the total field in $$\V_0 \backslash V_1\$$ is $$\0\$$.

Setup 2

Now suppose that in addition to the first object, we introduce another object which occupies volume $$\V_2\$$ and is bounded by surface $$\S_2\$$, and has the same material properties as the first object $$\(\epsilon, \mu)\$$.

I know that the equivalence principle can be applied individually to each object one at a time, to study the fields generated inside them.

However, does anything prevent me from treating these objects as one, i.e. $$\V_1 \cup V_2\$$ and $$\S_1 \cup S_2\$$? In other words, can I simultaneously introduce equivalent currents which exist on $$\S_1\$$ and equivalent currents that exist on $$\S_2\$$, which collectively generate the correct fields inside both objects, and collectively generate a $$\0\$$ field in $$\V_0 \backslash (V_1 \cup V_2)\$$?

I couldn't find any clear references to such a situation online; any insight would be appreciated. Thanks!