The best replacement for stirrups is roper stirrups.

The best alternatives to stirrups are rated as following:
★★★★★  roper stirrups
★★★★★  crooked stirrups
★  ties

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Speaking about stirrups, one probably shouldn't forget about sanitary socks.

Other search terms related to stirrups and their ratings are:
★★★★★ sanitary socks
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stirrups
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