In "We watched Obama speak," what is the technical reason for it not being "We watched Obama spoke"?
The second verb here is the bare infinitive.
Edit: a participle could also be used, i.e we watched Obama speaking. The difference in meaning would be aspectual. The first implies that his speaking was a single event (which was watched in full) The second implies that it was a process (which may or may not have been watched in full).
The technical reason is that the object complement clause of We watched, viz.
(for) Obama (to) speak
is an Infinitive object complement and, since infinitives are tenseless, it can't take a past tense form like spoke. Sense verbs like see, watch, listen, etc. often use infinitive complements without the infinitive complementizer to.