can you just "pick up a fake passport"?

Good question. I certainly know nothing about it.

But Thailand has (or had ) a reputation for being a place to do so:

http://www.scmp.com/news/asia/southeast ... t-passport

https://www.theguardian.com/world/2014/ ... angs-world

http://www.reuters.com/article/us-malay ... KD20140310

Interpol issues red notice for Scottish teacher wanted for Myanmar murder
Friday 14 April 2017 09.38 BST
Interpol photo
Interpol has issued a red notice for a Scottish teacher wanted by police over the suspected murder of a colleague in Myanmar.
Harris Binotti, 26, fled the south-east Asian country in November last year, hours before Gary Ferguson was found dead.

Binotti, from Dumfries, has been at the centre of an international manhunt since Ferguson was found with fatal head and chest wounds five months ago.

The Interpol notice, which gives the teacher’s full name as Harris John Italo Binotti, says he is wanted on a charge of murder. It describes him as having brown hair and blue eyes and measuring about 1.68 metres (5ft 5in).
Red notices are issued to international police forces when the suspect is wanted for prosecution or to serve a sentence based on an arrest warrant or court decision.

Ferguson’s brother Martin said the family were relieved the red notice had finally been issued.
He wrote on his Facebook page: “After such a hard and long fight, I am in tears to announce that Binotti has finally been put on the Interpol list of wanted persons. He can now be arrested internationally.”
https://www.theguardian.com/world/2017/ ... or-myanmar

Interpol Wanted Notice.