Back to back tee boxes. Who shoots first?

This is probably more of an etiquette than a rule thing but I've only been playing three years and today is the first time I came across this situation. We were teeing off on the 10th but the tee box runs back to back with the 14th a mere 10 yards between the two. We arrived at the 10th while another group arrived at the 14th. I gave the courtesy 'nod' but got nothing from them, whatever, their problem not mine but communication was obviously not imminent. Question: Is there a rule that dictates which group shoots first when the tee boxes are close enough so as to cause a distraction?

On my home course it's the 11th and 13th that set up like that.  We usually just alternate.  Someone tees off from one, then from the other, with the next guy in line teeing his ball and getting ready while someone is hitting from the other tee.  It means that neither group has to wait excessively.

It's happened to me a few times. What I did was ask the other golfers on the adjacent hole if they wanted to go first. If they said "no" then me and my group  tee'd off. If they said "yes" we let them have the honors. Ift hey said nothing then I would start our group teeing off. What neither group did was leave the tee box area until the other group was finished teeing off. alot of times we just started conversing with each other untill everyone had hit their shots.

Now on one course I played on, up in Washington State, there was a sign posted that gave the honors to one group over the other if they arrived at the same time. The lower numbered hole tee'd off first.

Your example is 4 holes apart, so my answer doesn't really help you much but ... my personal "rule" is that I always try to make sure that the group ahead plays first.  For example, I slice one onto an adjacent fairway and I get to near my ball at a similar time to a group on the tee.  If I'm playing up 6, and the adjacent hole is 7, then I'd like them to all hit their tee shots before I mosey on out to my ball.  If I try to beat them to it, and hold them up a bit, then I'm also only potentially holding my group up as well.  OTOH, if they're on 6 and and I'm on 7, then I want to go first and get out of their way.

That is interesting.  I would have thought that for Pace of Play reasons they would have the higher number hole tee off first.

I agree that this makes the most sense.  At the Old Course, holes 7 and 11 cross, and the players on hole 11 have priority - presumably because they are further into their round.

Years ago, the Fort Sill Golf Club course in Oklahoma had a similar situation. The No. 12 and No. 14 tee boxes were side by side, separated by a strip of rough.

On occasion, if both tee markers were in the back, two groups hitting at the same time could endanger each other. You just had to coordinate.