After hours of reading threads and following guides here and there, the definitive guide that got the driver working was here. Following the steps listed in that post:
$ # install necessary software
$ sudo apt-get update
$ sudo apt-get install bc git
$ # download raspbian kernel sources, takes some minutes
$ sudo wget "https://raw.githubusercontent.com/notro/rpi-source/master/rpi-source" -O /usr/bin/rpi-source
$ sudo chmod 755 /usr/bin/rpi-source
$ # download the rtl8812au kernel driver and compile it, takes some minutes
$ git clone "https://github.com/gnab/rtl8812au"
$ cd rtl8812au
$ sed -i 's/CONFIG_PLATFORM_I386_PC = y/CONFIG_PLATFORM_I386_PC = n/g' Makefile
$ sed -i 's/CONFIG_PLATFORM_ARM_RPI = n/CONFIG_PLATFORM_ARM_RPI = y/g' Makefile
$ # copy the driver and use it
$ sudo cp 8812au.ko /lib/modules/`uname -r`/kernel/drivers/net/wireless
$ sudo depmod -a
$ sudo modprobe 8812au
$ # check wlan0 interface appeared
The last comment says
#check wlan0 interface appeared but if you still have the RPi on-board WiFi activated, the dongle will appear as
wlan1. I recommend that you disable the on-board WiFi chip in order to save power and avoid causing interference if you'll be using a 2.4Ghz band.
These steps worked out flawlessly and soon after and I was connected to an ASUS AC1900 (802.11ac dual-band 5Ghz) router. As of the latest edit to this post I have been using the dongle for about 2 days for about 10 hours a day without problems.
There are many posts by MrEngman on the Raspberry Pi forums answering questions related to the Realtek WiFi chipset drivers and you'll encounter many 404'd Dropbox links to drivers and install-wifi shell scripts attributed to MrEngman. I didn't follow any of them as the information was too fragmented and he does not seem active anymore. When following any of his tips or hints, be sure to go to his site first and follow the information there; don't go forum diving in old threads. Especially when there's different drivers, kernels and versions at play.