What is the best routing protocol for a large company that has international branches?

There are many reasons to switch from static routing to a routing protocol. The main reason that I’ve seen is the ease of adding a new subnet to a network. If you were to add a new subnet in a static routing environment, you would need to add that static route to every layer 3 device. That is a time consuming and tedious process. With a routing protocol, you would just add that subnet to the device running the routing protocol that is closest to where that subnet exists and it will propagate through the network. The routing protocol will use its metrics to determine what the best route to get there is. This means that you don’t have to touch every layer 3 device and add the static route.

For External Routes I Vote (BGP)

For internal :- I prefare OSPF 

For the WAN part:  I vote for BGP as it has multiple attribute to control the traffic.

For the LAN part (internal branch): I vote for OSPF or other IGP routing protocol as it has faster convergency within local site.