In a burst of enthusiasm when I was much younger, I bought a large reel
of solder which I barely used.

It must have been at least 20 or more years ago. It looks similar to
this:
http://img-europe.electrocomponents....C557130-01.jpg

Now I need to do some soldering again. Would that old 60/40 multicore
of mine have deteriorated?

I'm thinking the rosin in the multicores may have gone off in some way
because when I tin the tip and then leave it unused for 10 minutes, it
turns brown and uneven on the tip. I'm using brand new tips.

Maybe this is normal and happens even with less old 60/40 solder?