This question is related to this one: Which limits are preserved by a reflective left-adjoint?

(And in fact may be seen as a special case, but I think it merits its own question).

Suppose that $C$ is a reflective subcategory of $D$ and that coproducts in $D$ commute with finite limits. Notice that coproducts computed in $C$ are not necessarily the same as computing them in $D$; one may have to apply the reflector afterwards. Does it however follow that coproducts in $C$ commute with finite limits? **Edit: No. See Todd's answer below.** If not, under what assumptions will this be nonetheless true?

**Edit**: Obviously this is true if the reflector is left-exact. I am interested in conditions weaker than this.

**Subquestion**: Is there anything that can be said in the special situation where $C$ is the category of free-algebras for a monad on $D$ and the adjunction is the obvious one?