Sore achilles tendon after first use of clipless pedals

Another thought: soreness of the achilles tendon can be caused by the seat being too low, have your new shoes/pedals changed your leg extension slightly?

You may have thought of this, but is this the first time you have used the shoes? Could it simply be the shoe rubbing your tendon?

I would really recommend getting a bike fit. I had one and moving my cleats into what felt like the wrong position cured 20 years of knee pain

Fore and aft position of cleats, changes the strain on heel achilles. Also see if your toes are facing down slightly whilst riding, this will help with heel strain. The adjustment can be very small but make huge difference in comfort.

My own example is that my summer shoes are fine and I don't suffer with heel pain, but my winter boots are slight thicker in the sole and I get minor heel discomfort.

Agree with the previous comment.

You may need to experiment with fore and aft position of the cleat as well as the angle parallel to the direction of travel, until you find the most comfortable position for you.

I also use cleats that allow a degree of float, I find that little bit of movement helps with comfort.

If you just can't resolve the problem, then a proper fitting might help you get the setup right.

I moved to clipless for the first time in the summer. First few rides I found my knees getting sore- for the first time ever riding. I have the A530 flat on one side/ clips on the other pedals, so looked very carefully at my foot position clipped in/ on the flat side. There was only a few degrees in it, but my toes were pointing out slightly more when on the flat side. Adjusted the angle of the cleats slightly, to allow this- and, voila, no more pain at all.

So, purely on the basis of my own experience, I would suggest looking very carefully at the position of the cleats, both fore and aft, and the angle, to replicate as closely as possible the position of your foot on the pedals that you are used to.