definitions of extensive and strategic (normal) games are very different. Here is the question, what would you call a game which is extensive but in each step strategic.
For instance at each step both players choose to play or not simultaneously and each step is also affected by previous steps.
If someone has seen such a thing in any paper or textbook, please refer to it.
On
In the book Game Theory by Fudenberg and Tirole, they call such games multi-stage games with observed actions. These are games where there are rounds (finitely or infinitely many), and in each round player simultaneously choose some action which is observed afterwards. PAyoffs may depend in any way on the whole sequence of action profiles constituting a play.
Special cases of such multi-stage games with observed actions are repeated games (there are extensions that do not fit into this framework) and stochastic games (at least as far as their game tree is concerned).
Osborne and Rubinstein have an advanced textbook that deals with these issues, such as repeated games and representing simultaneous moves in extensive form. http://books.osborne.economics.utoronto.ca/ I also like Jim Friedman's books, and he was an early master of repeated games. http://www.amazon.com/Theory-Applications-Economics-James-Friedman/dp/0195036603