What are the rules for this grammatical construction: 我去听他的演讲去晚了 I'm going through an anki deck and have come across the following sentence: 我去听他的演讲去晚了

The verb is mentioned twice, the first time to describe the goal (听他的演讲) and secondly to describe the verb, like an adverb (she went late). 

NOTE: I've only ever seen adverbs using 地 or 得 

Can I do this with any verb?  e.g. 我吃古老肉吃快了？

What about two character verbs?
Two clauses compressed to one by reintroducing the verb. The first instance is descriptive (I am going to listen to his lecture). The second instance describes the result of the action (I came late).

You need to reintroduce the verb to give 晚 the proper context. Without it, 晚 would refer to 演讲.

Sometimes you can omit one instance of the verb: 你（说）汉语说得很流利。

Sometimes you really have no choice: 你说汉语比我说汉语说得好。