We introduce a class of hybrid marked point processes, which encompasses and
extends continuous-time Markov chains and Hawkes processes. While this flexible
class amalgamates such existing processes, it also contains novel processes
with complex dynamics. These processes are defined implicitly via their
intensity and are endowed with a state process that interacts with
past-dependent events. The key example we entertain is an extension of a Hawkes
process, a state-dependent Hawkes process interacting with its state process.
We show the existence and uniqueness of hybrid marked point processes under
general assumptions, extending the results of Massouli\'e (1998) on interacting
point processes.