The general answer is "yes" but I have no idea how you've got everything wired.
#1 - what do you have for an "ignition" switch?
If when turned "on" it connects your battery-
positive to your coil - it will run regardless of what the alternator is doing.
Considering that once "off" and you try to restart - and you've got no spark - it sounds obvious that there is no 12 volts-positive wired to your coil from your ignition switch. It DOES sound like you've got a connection between your alternator and your ignition. Since you've got it hooked up "two wire" I assume and not "three wire", the field-exciter circuit on the alternator is probably backfeeding to your ignition circuit when charging and keeping it running. Just a guess since I have no idea how you've got it wired. The exciter wire goes to the plug-in connection post #1.
By the way - there's a subtle difference between "two wire" and "three wire." Two-wire is when you loop the small voltage-sensing wire (#2 at the plug-in connectors) back to the big post on the alternator. Three-wire is when you run that voltage sensing wire all the way to the battery. It serves no purpose on a tractor since the battery is often close. It IS used on autos because the voltage at the battery tends to be slightly different then at the alternator output post.