Build trust for federated search between two SharePoint Server farms

Federated search is when you aim to receive search result from separate SharePoint (on-premises) by performing a search query in a separate on-premise SharePoint farm.
If you have done such configuration probably you have seen the official documentation for setting it up. This procedure will work in most of the cases.
However, this will not work if you do not have outbound connectivity from the remote farm that will receive the search query (ReceivingFarm) to the farm that is sending the query (SendingFarm).
In that case the federated search will be possible as long as the SendingFarm can access the ReceivingFarm, vice versa is not required, but you should take a bit different approach when building the trust since the SendingFarm web app metadata end point will not be available.
The first thing that needs to be done is to export the root and the token signing certificates from the SendingFarm and also get the Issuer Name (NameIdentifier) of the SendingFarm STS .

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## Export Root Certificate
$rootCert = (Get-SPCertificateAuthority).RootCertificate
$rootCert.Export("Cert") | Set-Content "C:\SendingFarmRoot.cer" -Encoding byte
 
## Export Signing Certificate
$stsCert = (Get-SPSecurityTokenServiceConfig).LocalLoginProvider.SigningCertificate
$stsCert.Export("Cert") | Set-Content "C:\SendingFarmSTS.cer" -Encoding byte
 
## Get the STS Issuer Name
$issuerName = (Get-SPSecurityTokenServiceConfig).NameIdentifier

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The difference from the official procedure will be how we are going to create the trusted token issuer and the trusted root authority in the ReceivingFarm, this is step 3 in the official procedure.
First copy the SendingFarm certificated to the ReceivingFarm.
Having above done you can create the trusted security token issuer and the trusted root authority  in the ReceivingFarm.

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## Read SendingFarm Signing certificate
$stsCert = Get-PfxCertificate "C:\Install\Certs\SendingFarmSTS.cer"
## Read SendingFarm root certificate
$rootCert = Get-PfxCertificate "C:\Install\Certs\SendingFarmRoot.cer"
# Create a trusted security token issuer
$i = New-SPTrustedSecurityTokenIssuer -Name "SendingFarm" `
                                      -Certificate $stsCert `
                                      -IsTrustBroker:$false `
                                      -RegisteredIssuerName "<SendingFarm IssuerName>"
 
# Configure trust of the token-signing certificate'
# by adding the trust used to sign oAuth tokens'
# to the list of trusted root authorities'
# in ReceivingFarm
New-SPTrustedRootAuthority -Name "SendingFarm" `
                           -Certificate $rootCert

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Now, you can continue with the trust configuration as it is described in the documentation.

I hope you found this helpful!