Verified answer

Your grammar book is right, the proper French sentence is " Il a couru après lui" .

" Il lui a couru après" is exactly what they say, a colloquial form frequently used, but not correct in formal French .

The use of "lui" without a preposition and before the verb is limited to cases when "lui" is an indirect object pronoun, i.e. when the omitted preposition is either "à" or "de" .

When "lui" is an adverbal complement used with other prepositions ( pour, comme, avec, devant, avant,derrière, après,...), it must stay after the verb and keep its preposition .

So, for a formal writing you should use the proper form : " Il a couru après lui" .

Kay I totally agree with you "Il lui a couru après" (he ran after him) is excellent French the prob with the other expression is that to me - a native franco- is a bit more confuse and would require more context for this reason your first bet was the right one. Keep on the good work!