When your contract is in the phase before it is signed, all inventory items added to the contract will automatically be sent to fulfillment.

After your contract is signed, changes are no longer automatically sent to fulfillment. After editing a signed contract, you will need to either turn fulfillment sync on, manually mark the contract as signed again, or have the client re-sign the contract to have the changes take place in fulfillment

Note: Anything in the fulfillment tab is completely internal only and does not affect the contract. 

The contract affects the fulfillment, not the other way around

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