Is there a phrase that Japanese people use to express amusement when seeing
each other unexpectedly? For example, if you just said goodbye and then ran
into each other somewhere else?
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expressions set-phrases greetings
"fancy meeting you here"
Probably most commonly heard is あらまあ (shortened to あら). Although it rather
expresses surprise, it fits into the scenario:

>
> [あらまあ](http://dictionary.goo.ne.jp/jn/7309/meaning/m0u/%E3%81%82%E3%82%89%E3%81%BE/)
>
> ［感］軽い驚きや思いがけなさを表すときに発する語。多く女性が使う。「―、どうしたの」
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A common word for this is [奇遇【きぐう】](http://jisho.org/word/%E5%A5%87%E9%81%87)
(not to be confused with 奇偶: oddness and evenness (of numbers)). 奇遇ですね is an
idiomatic phrase worth memorizing as-is.

>   * こんなところで会うとは奇遇ですね。 What a coincidence to meet you here!
>   * 君も北海道生まれか、奇遇だね。 Oh, you were born in Hokkaido, too? What a coincidence.
>

But if you just said goodbye and then ran into each other soon after that, you
can't use this phrase. (Can you say "fancy meeting you here" in that
situation?)
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2016-08-04T09:59:42.693
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If I meet someone in an unexpected place in an unexpected time, I would say;

これは、これは - wow, Jesus!

(こんな場所でお会いするとは)これはまた、奇遇(偶然)ですね - It's really coincidence (to meet you in such
a place)

おや（あら）、まあ - Oh My God!

おや（あら）、お珍しい - Wow. I'm surprised (to meet you in such place).

いや、懐かしい(casual), Dear! Long time no see.

お懐かしゅうございます - Ditto, but polite, or said to senior).

If I ran into someone whom I just said “Goodbye” somewhere else, I would say;

おや、また会い(お会いし)ましたね。- Oh. We saw again.
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