From the eBay listing:
Remanufactured
A properly rebuilt automotive part. The item has been completely disassembled, cleaned, and examined for wear and breakage. Worn out, missing or non-functioning components have been replaced with new or rebuilt components. It is the functional equivalent of a new part and is virtually indistinguishable from a new part. See the seller's listing for full details.But, in the Q/A in the listing:
[i]Questions and answers about this item
Q: Normally there is a diaphram with a capulary tube on this valve;is this a file photo or is the valve incomlete ???
A: This valve does not have a capillary tube. It controls the hot water flow like most vechcles of later years---- by turning the knob and rod under the dash on the passenger side of the cab. Thanks for the interest. Jim
This is not remanufactured in terms of his own definition however, in the Q/A he covers his inadequately remanufactured part.
Here is a correct valve. (as you said, you'd need a core)
Your choices are:
1. Ask to return the item (for full refund - maybe paying for shipping one-way or both-ways).
2. Keep the item and either use it or get a different one. Jim Carter sells one at double the price, if you do not have a core.
In my opinion, the seller (phlobe47) is wrong. However, he will say that he clearly stated the capillary tube was missing.