The following paragraph is meant as a high level summary of the issue that is occurring, and I attempt to explain the relevant parts in more detail in the subsequent paragraphs.
Immediately after upgrading from macOS 10.12.3 to 10.12.4 I noticed that when my MacBook Pro (Late 2013) boots up or wakes while plugged into my two external displays via Mini DisplayPort, my MacBook has extreme difficulty getting them both working at the same time. The MacBook's screen will turn on and off as it tries to get in sync with the external displays (to start displaying on them as normal). Meanwhile, the displays seem to cycle between receiving a signal and receiving no signal as suggested by the fact that the displays will wake, but then display "No Signal" and go back into power-save mode. Eventually, after many cycles of this, one or both displays may "get in sync" with the MacBook and function normally from there on out. But sometimes it goes through many such cycles without getting in sync. When the displays and the MacBook fail to get in sync after many cycles and continue to cycle as described, I may unplug one or both external monitors and try to get them to sync up individually, which is usually easier than trying both at once. This makes sense because, as I explain below, this problem is effectively a race condition and appears to be compounded with every additional external display in use. Finally, the more serious case is when the cycle I've described ends in the MacBook entering an unresponsive state and requiring a forced shutdown which can lead to loss of data or unsaved work. Furthermore, this whole issue presents itself every time I boot up or wake the MacBook.
To be clear, the displays wake from power-save mode upon receiving a signal from my MacBook (e.g. when it boots or wakes from sleep). But after receiving that initial signal waking the display from power-save mode, the display shows "No Signal" and goes back into power-save mode. Meanwhile, the display on my MacBook goes from normal to a blank screen for a couple seconds and back to normal as it seemingly switches between outputting to the external display and not outputting. This is the cycle that continuously repeats until either each respective monitor gets in sync with the MacBook and works as normal, or the MacBook enters an unresponsive state as described below. In the case where the monitors do eventually get in sync, it appears to occur the moment the monitor waking happens to coincide with the MacBook outputting a consistent signal to that respective monitor. So, it's effectively a race condition. The reason I say the MacBook cycles between outputting to the monitor and not outputting to it is because, while it's in this cycle, I'll observe windows I had open on the external monitor prior to the MacBook sleeping toggle between being displayed on the MacBook's display and not being displayed at all (presumably they're being output to the external display).
As I mentioned above, there is an even more troublesome situation I've now encountered multiple times where, upon waking from sleep, the MacBook will enter the cycle of trying to get in sync with the monitors for a while and then enter an unresponsive state, showing a blank screen. Plugging in and unplugging the monitors from the MacBook or power cycling the monitors seems to have no effect in this situation. Even if the MacBook is recognizing one of the monitors while in this unresponsive state, that monitor is also just rendering a blank screen (but is not going into sleep mode therefore is receiving a signal). My only option has been to force shut down my MacBook after this has happened by pressing the power button and leaving it pressed.
I have two Acer S241HL displays each connected via Mini DisplayPort. I had never had this issue before and it started immediately after I upgraded from 10.12.3 to 10.12.4. In the past, my MacBook's screen might turn on/off once or twice when booting up (and maybe waking?) before all displays were on and rendering properly. But that was always quick and never an issue. This is a much greater extreme and a major inconvenience as I have to wait and sometimes mess with the monitors every time my MacBook wakes or boots up. Even more serious is that it is now dangerous for me to wake my MacBook if I have monitors plugged in due to the risk of it entering an unresponsive state, thus requiring a forced shut down. This can easily cause problems including loss of data and unsaved work.
I've contacted Apple Support to escalate this issue to engineering. They should be aware of this thread, so if you experience this same issue, please comment with details of your setup such as which video out port(s) you're using and what kinds of cables/adapters, as well as any other pertinent information.