[jpstewart]

I think I see the problem.  The screenshot shows the system looking in /usr/lib64 and other "lib64" directories, which means the program requires a 64-bit version of the library.  The Nekoware package only has an N32 library, not a 64-bit one.

My Octane has a 64-bit version of libz.so in /usr/lib64 which means it came from the base IRIX installation (although the 64-bit portion may have been optional at installation time).  There's another in /usr/freeware/lib64 which is somewhat newer but not in a path that will be searched by default.

My recommendation is to remove the Nekoware zlib package since it won't help in this situation.  Then either install the 64-bit library to /usr/lib64 from the IRIX installation media, or install the freeware package and set LD_LIBRARY64_PATH to point to it.