Double-sided Eyelets?
The double-sided ones are often called &quot;grommets&quot;.  Good search terms are &quot;fabric grommets&quot; or &quot;fabric grommet tool&quot;.  Amazon has a huge selection, including different sizes and colors.
The grommets consist of two pieces.  One is like a shaped washer that goes on the back. The other piece is similar to the eyelets you're using, but it, and the installation tool, are designed to stretch and flare the eyelet tube instead of splitting it into a &quot;star&quot;.  Once it's installed, it looks fine on both sides.