Returning Related Record Fields via SOAP WS Search

In short, I am trying to use a Transaction Saved Search (Vendor Bill) to find the related Purchase Order lines, to validate the Billed Line Amount vs the PO Line Amount. We are using the POs to estimate a cost, but what we care about is the Billed cost in the end.

When I set up a saved search (going to show pictures below), I can find the lines from both sides no problem. I constrain on Vendor Bill, and use the 'Applied To Transaction' fields to pull in the PO line to find mismatches. However, in the 3 years of using SOAP Web Services, I have never been able to figure out how to get the same fields from any type of search. The only thing that comes back is the internal id of the Applied To Transaction, not the individual fields I specify on the saved search.