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When first studying Japanese, I learned that "after" in Japanese is 後で. But asking around and looking at regular usage, a lot of Japanese will tell me a sentence like the following is slightly unnatural:

映画を見た後で家に帰ってきました。

I actually tried making a few sentences with 後で and I kept getting corrected to either drop the で or use 後に. So, is 後で being used less these days? I see plenty of examples being used in the Tanaka Corpus, but that's a bit old.

There are several things to consider in answering your question.

An objective answer to your question would require a direct comparison between the total usage of 後で at two different points in time. This is not a trivial task and it requires more specific information, i.e. exactly which two time points to compare, what corpora you will base your conclusions on, whether you want to include Nounの後で or other variants as well, etc. You could then see if there is a pattern of attrition and perhaps even investigate the reasons.

If you are just looking for a subjective answer, then it is reasonable to ask native speakers but I think you would also have to use example sentences generated by native speakers. For example you could take 20 sentences containing 後で from a corpus and show them to Japanese people to ask whether they feel something is unnatural in the sentence. The reason I believe it is important to avoid non-native speaker examples is that they often have errors which can make the sentence sound unnatural for a variety of reasons that might not be immediately apparent, even to native speaker.

Take your sample sentence, for example.

映画を見た後で家に帰ってきました。

According to the Dictionary of Basic Japanese Grammar, 後で indicates "any space of time after" whereas ～てから indicates "the space of time following after". It may well be the case that 　　

映画を見てから家に帰ってきました。

would sound more natural to a native speaker precisely because of the implication that the action of going home immediately followed the action of watching the movie, and was not separated by "any space of time". So you could argue that 後で is not the appropriate term to use in this specific circumstance because of the meaning of the sentence. You have to be careful attributing a cause to an effect and so I think it is best to avoid sentences made by non native speakers.

Finally, when something does end up becoming obsolete there is usually a reason for it, and it's often the case that it is superseded by something else. You might investigate which terms are more likely to be used instead of 後で as another way to explore your question.

So without directly answering your question, I hope that helped. :)

This sentence sounds unnatural indeed. As you were instructed, either drop the で or use 後に, that will make it more natural.

I am quite sure the sentence has never sounded natural, actually. The reason it sounds unnatural is not because the language is changing.

In particular, 後で is used all the time, you can't spend a day at school or work without hearing it at least once.

It is just that you can't put this 後で inside this sentence. Consider this English sentence:

Afterwards watching the movie I came back home.

It sounds weird, not because "afterwards" is fading in usage, but just because it is the wrong word here, you should use "after" instead.