Spigot leaks when sprinklers come on
Sounds like an anti-siphon reaction to me, full water pressure in the hose going to your roof, and the house water pressure temporarily drops from the large demand from those two zones until the sprinklers are fully charged. You can try adding an anti-siphon or backflow prevention valve on the faucet to verify (they make valves that screw on between the faucet and hose). If the leaking stops or significantly reduces, then you know that's the cause.

As for why it's leaking out the stem, I'm not sure, but it's possible that's by design. The cap on top should be a vacuum breaker, allowing air in the lines after the water is shutoff and pressure drops.