Is there a TREP API call to obtain search results like these displayed by EIKON?
Eikon utilize data from many sources. As far as I know, APIs that may provide this kind of data, it could be Knowledge Direct (TRKD) and DataScope Select (DSS)
You may contact Thomson Reuters Account Team if you want to access the APIs.
Basically TREP API provides access to the capabilities and content of the Thomson Reuters Elektron Realtime Feeds, which include the ability to consume Elektron feed and/or 3rd party feeds data via TREP.
TREP API does not support functionality like the Eikon Search you mentioned. Comparing the functionality of TREP API with Eikon Desktop, it should be the same as Eikon Quote component which was designed for request Real-Time data. As far as I understand, the Eikon search does not return Real-Time data. It seems to returns some Referenece or Fundamental data instead.
Of course, it doesn't have to be sourced from the Elektron feed, but if it's visible to the EIKON user, then there's an API and connectivity to TR which is capable of obtaining that data. What is it? Can it be used by an app that I write?