How to solve the TIBCO Silver Fabric issue: engine stuck on status 'allocating'

After installing the 64-bit distributions of TIBCO Silver Fabric on my Windows 7 machine, I encountered an error when I started up a Stack containing components Administrator and BusinessWorks (from the TIBCO Silver Fabric Enabler for ActiveMatrix BusinessWorks 2.6.0). The engine where Silver Fabric is trying to run Administrator is stuck on status "allocating".

This is an excerpt from the engine logging:
10/13/14 07:39:24.994 INFO: [ProcessWrapper] [STDERR] TibrvException[error=901,message=Library not found: tibrvj]
10/13/14 07:39:24.994 INFO: [ProcessWrapper] [STDERR] at com.tibco.tibrv.Tibrv.loadLib(Tibrv.java:476)
10/13/14 07:39:24.994 INFO: [ProcessWrapper] [STDERR] at com.tibco.tibrv.Tibrv.open(Tibrv.java:275) 10/13/14 07:39:25.009 INFO: [ProcessWrapper] [STDERR] TibrvException[error=27,message=Native implementation required]
10/13/14 07:39:25.025 INFO: [ProcessWrapper] [STDERR] at COM.TIBCO.hawk.console.nest.shared.TibrvProxy.open(TibrvProxy.java:33)
10/13/14 07:39:25.025 INFO: [ProcessWrapper] [STDERR] at com.tibco.tibrv.TibrvRvdTransport.init(TibrvRvdTransport.java:67)
10/13/14 07:39:25.041 INFO: [ProcessWrapper] [STDERR] at COM.TIBCO.hawk.console.nest.shared.TibrvProxy.open(TibrvProxy.java:24)
10/13/14 07:39:25.041 INFO: [ProcessWrapper] [STDERR] at com.tibco.tibrv.TibrvRvdTransport.(TibrvRvdTransport.java:50)
10/13/14 07:39:25.041 INFO: [ProcessWrapper] [STDERR] at COM.TIBCO.hawk.agent.AgentTibrvTransports.initTransports(AgentTibrvTransports.java:75)

At first, I thought that the RV_HOME system variable wasn't setup properly because of the "Native implementation required" message in the logs. To investigate this, I found an article that explained how to test if the tibrvj library could load propery. After executing the JAR file, the following exception returned:
Exception in thread "main" java.lang.UnsatisfiedLinkError: C:\TIBCO\tibrv\8.2\bin\tibrvj.dll: Can't load AMD 64-bit .dll on a IA 32-bit platform

Usually this means that correct C++ runtime components are missing. After installing the correct Microsoft Visual C++ Redistributable Package version, the problem was solved!