In the absolute there is an inconsistency in this sentence, the reason being that "dès que" is the indication of a point in time. It's made flagrant by an amplification.
- L'hymne sera diffusé dès que le vaisseau fera son trajet de la Terre à Mars. (Le trajet prend quelque chose comme 200 jours.)
As from the first to the last of the 200 odd days the spaceship is accomplishing its trip, we have a multitude of points of reference, we don't understand.
You can only specify an action that occurs at an absolute point in time or a point idealised as such, or the end of an action. There is not much of a margin for diverging from this absolute.
- Dès que l'engin aura explosé un gaz nocif commencera à se diffuser dans l'atmosphère.
In the example above, although an explosion is rather a point action by most standards, it is better to use "aura explosé" than "explosera", although this latter choice is acceptable.
As the action "finir de manger" is not at all of the sort that occurs at an absolute point in time but is an action that takes definitely some time, for example in a meal it could be the time taken for the dessert, the inconsistency of the formulation is also present, although one might not feel that as sharply because of the verb and the context.
If we consider a point action simultaneity is the norm but only in that case.
- L'explosion sera amorcée dès que le contact sera fait.
Here the use of the future in both clauses sounds right.
The start of the explosion and the coming into existence of the contact are not yet perfectly simultaneous but for most practical purposes they can be considered to be.
There are however perfect examples but they imply point actions or states since "dès que" implies a point in time; one finds them for instance in the domain of astronomy where the realisation of certain angular measures correspond to simultaneous phenomena.
All of this indicates that either you must say
"Il parlera dès qu'il aura fini de manger."
because "avoir fini de manger" describes a state and not an action as does on the contrary "finir de manger" and there is no simultaneity of action, or you must say
"Il parlera pendant qu'il finira de manger.",
and in this case there is simultaneity.