How do I "rubber band" my users when they click an external link on my site?

there's no way to do this until you work with the affiliate merchand or any external site

I know it is possible because it was done on two of my Web sites a few years ago. But I have forgotten what was iinvolved.
I do not understand what you mean by " until you work with the affiliate merchand or any external site."   Of course that is when the access code is aailable.  I was not askiing about that but the proceess involved (i.e. what needs to be written )

when you click any link on your site, make sure that you set it to open in a new window (tab) by adding target="_blank"

for example: <a href="https://www.youtube.com/channel/UC3K0LygH48gjj6RaV22ZRxQ" target="_blank">Visit <b>Les Pieds Gelés</b> channel on <b>YouTube<b/></a>

As already mentioned, the only way someone can be sent back to your site, after they've taken an action on an affiliate (3rd party) site, is if that affiliate site voluntarily redirects them back to your site.

You would need to collaborate with the affiliate site for this to be possible. Basically you would have a relationship with the affiliate with a setup to the effect of:

1.) Your affiliate-URL (the external link) is used to recognize "you" sent the visitor to the affiliate
2.) The affiliate agrees that anytime a visitor is sent through this aff-URL, when a purchase is completed, they will redirect the visitor back to your site (the redirect may be javascript based, but it doesn't matter). The affiliate would need to set this up on their side.

An example of this is using "paypal checkout" on your web site. You basically send a visitor into Paypal to make a payment. When the payment is completed, Paypal sends the visitor back to your site. However, you have no visibility into what the user is doing on the Paypal web site. You simply have an agreement that when the checkout event is complete, paypal sends the visitor back to your preferred URL (ex. thank-you-for-your-purchase.html).

This is something that would have to be collaboratively agreed to with the affiliate site. Because as previous comments have mentioned, this is impossible to do on your own. You have no ability to track the actions of a visitor, once they leave your site. This would be a massive security hole in all modern web browsers if you could monitor users actions when they are off-site. Hence, there is a trust-based agreement involved.

Thank you to Paul and Lucas.    
I have tow final follow up questions for each of you.
To Paul.  What will that bit of code achieve in the light of the ccomments from Lucas:

To Lucas.   I take all the points from your comprehensive response.   I do understand that I could not track the movements by a uuser on the target site.   Nor would I wish to do so.

I was seeking to make it easier for the user to return to my site to make a different choice without having to log in again
or keep hitting the back button after a long session on the initial target sit.
Further i will not be sending users only to affiliate target sites but also points of reference for information.

I would appreciate your further observations please.

Thanks for your help.

Paul's information is helpful based on your response. For all of the aff-URLs, if you target them to open a new window, that means your site will be present in the previous window. So if someone finishes shopping on the affiliate site, then they close their browser, your site will be sitting there in the underlying window.

@shaun, the thing about iframes is that if you try to load a third party site and that site is https enabled, it usually will fail to load or it will throw security errors. I believe that's default functionality in most browsers. I could see how maybe 5-10 years ago, that could have been the exact solution to this request though.

I merely meant that the previous sites that OP mentioned might have been done with IFrames, not suggesting it as a solution. Some site prevent IFrames

Yeah solution isn't what I meant. I'm guessing that is what may have been used back when he saw it working years ago.

that was impressive deduction because it was about six years ago.  Lucas you mention "your site will be sitting there in the underlying window. "1.    Not sure what this means.   2.   Does this relate to your earlier point about having to work with the Affiliate to make this happen.   3.  What about  my user going to an external information site to access more information from my link.   Would I hve to get the co-opertion of the information site in the same way as I would the Affiliate.

Thanks for all your help.