Yes, a participial phrase can be used along with a perfect-tense verb in the manner shown in this sentence that you presented: “Working in cooperation with three midwives and one theatre nurse, I have successfully carried out a surgical operation on a pregnant woman with history of obstructed labour.” The crucial consideration in such constructions is that the action in the main clause—“carry out” in this particular case—has just been completed at the moment of speaking, thus requiring the present perfect tense “have carried out.” Of course, if the action was carried out at a definite time in the past, the simple perfect tense will be called for: “Working in cooperation with three midwives and one theatre nurse, I successfully carried out a surgical operation last week on a pregnant woman with history of obstructed labour.” If the action was carried out at some indefinite time in the past, the past perfect tense will be it: “Working in cooperation with three midwives and one theatre nurse, I had successfully carried out a surgical operation on a pregnant woman with history of obstructed labour.”
Just keep in mind that the participial phrase in such sentences serves as an adverbial modifier of the main clause, and can function as such without any modification no matter what tense or modality the verb of the main clause may take, as in this future-tense sentence, “Working in cooperation with three midwives and one theatre nurse, I will successfully carry out a surgical operation on a pregnant woman with history of obstructed labour,” or in this modal sentence, “Working in cooperation with three midwives and one theatre nurse, I can successfully carry out a surgical operation on a pregnant woman with history of obstructed labour.”
