Literally, 明後日の方向 (あさってのほうこう, asattenohoukou) means “the direction of the day after tomorrow”. This idiom metaphorically indicates in an ironic manner that something goes in a completely wrong and unpredictable direction. For instance, an uncoordinated person often throws a ball at 明後日の方向 (an unpredictable direction).
This expression can also be used figuratively to illustrate that something/someone goes far off the point. For instance, if one says “明後日の方向に組織改革をする” meaning “reform an institution in “明後日の方向”, that indicates that the reformation doesn’t go according to plan and goes off at a tangent.
明後日の方向 consists of the two words, “明後日” and “方向” with one particle, “の” between them. The first word means “the day after tomorrow”, and the latter means “direction”. Therefore, as mentioned already, the whole phrase “明後日の方向” denotes “the direction towards the day after tomorrow”.
“予想の斜め上を行く” is a synonym of 明後日の方向. Literally, it means “go up obliquely above one’s expectation”, and indicates that something is unexpected and off the wall. You can also use the word “想像” meaning “imagination” instead of “予想” meaning “expectation” used in this expression.
Comparison between 明後日の方向 and 予想の斜め上
Compared to 明後日の方向, 予想の斜め上を行く can be used for something either positive or negative, and it emphasizes that something is unique, bizarre and/or unexpected. On the other hand, 明後日の方向 usually stresses that something is unreasonable and off the point. Another difference is that 予想の斜め上を行く sounds more like a slang and is usually used on the Internet.
I kicked the ball and it flied 明後日の方向に (in the unexpected direction)
The discussion is developing 明後日の方向に (in the completely wrong direction).
The movie has become a big hit owing to its 予想の斜め上を行く展開 (storyline that goes up obliquely above my expectation => unique storyline )
I sent a LINE message to my friend and got 想像の斜め上を行く返事 (an unexpected reply)
Synonyms: 予想の斜め上を行く (see above)
Synonyms in English: unreasonable, unexpected
Related words and phrases: 想像 (imagination), 予想 (expectation)