In everyday conversation, 水を取り出して来てくれ does not make sense. But 水を持ってきてくれ makes sense. The first one literally means, (please) take water out from source and come back. For me, it does not work. That is the reason. The second one literally means, (please) bring water from source and come back. It would make sense.

You are wrong. The right way is the bellows. 1.Water is liquid.so we can't 取り出す water. 2.水を持って来てくれ→Not natural writing. 水を持ってきてくれ→Correct.at least it's more natural.

I think this is an interesting question, but it's not very easy to answer. So, let me try from my view. I'm sorry if I miss the point. (1)Basically, て connects two verbs and shows a sequence of actions like this: 夕べは、たくさん飲んで食べた。 I ate and drank a lot last night. (2)て also connects a main verb and an auxiliary verb like these: むずかしい本を読んでみた。 I tried to read a difficult book. ずっと日本語を勉強してきた。 I have been learning Japanese for long time. ともだちに会ってきた。 I saw my friend and came back (home). In this case, an auxiliary verb is written by ひらがな and it doesn't bear an original meaning or its orginal meaning weakens. みた from 読んでみた means to try, not to see. きた from 勉強してきた means to continue some action, not to come. きた from 会ってきた still means to come back, but it focuses more on main action=会う and it means, after finishing some action(=to see, in this case), to come back to an original location((ex)home, or office). Then, how about 取り出してきた? This has a nuance where someone pull something out of something/some place and come back. So, this still bears an original meaning 来る, but weakens. 古いアルバムを本棚の奥から取り出してきた。 She pulled out an old album from the book shelf (and came back). 古いアルバムを本棚の奥から持ってきた。 She brought an old album from the back of the book shelf (and came back). 取り出してきてくれ doesn't necessarily mean "GO and fetch something and come", but "fetch something and come". (If you ask someone in front of you to fetch something, he has to GO some place to take something AS RESULT. ) From the perspetive if these phrase has a meaning of "to go", 持ってきてくれ is the same as 取り出してきてくれ. I hope this won't confuse you.

Hello Andreas, Firstly, I agree on Answers. I would like to add a comment. 取り出す is used for something physically exist that you can grab. Therefore, 水を取り出す does not make sense because water cannot be grabbed. 冷蔵庫から水のボトルを取り出してきてくれ is understandable. But it might mislead because of the following reason. 取り出して + くる = put something out + come 持ってくる = bring something to someone (持って + くる = have + come) Because 取り出してくる can be divided into 2 words, people can understand that keeping something out and coming here (no necessity to bring it). Although I think usually they can understand what it means depending on the situation... So if you definitely want someone to bring something to you, you have to say ・・・を持ってきてくれ ・・・を取り出して持ってきてくれ（in case that 取り出して is needed） ではまた