Many years ago, China would not allow entry with a Taiwanese visa in your passport,
and Taiwan would not allow entry with a Chinese visa ( This was before China "opened up", and there were very few of us travelling there). What we did was arrange for a
second Canadian passport that was held at the Canadian High Commission in HKG - at
that time all flights in and out of China were via HKG - go to HKG for 2-3 days, exchange passports in HKG, apply for rush Chinese visa, go to China, return to HKG,
and then exchange the passports again for the originals.

Those were the good old days. I also recall having to get my Taiwanese visas in Seoul,
when what is now the Chinese embassy near the Chosun Hotel was then the Taiwanese
embassy. It was a great step forward when we were finally able to send our passports
to USA to get Taiwanese visas in advance, then finally there was a Taiwanese trade
office in Canada that was authorised to issue visas.