Basically any modern derailleur is compatible with friction shifting to a point. Try to stick with sub 10 speed shimano compatible derailleurs though (as some older friction shifters don’t pull enough cable to use all gears on sram ESP or other low pull ratio derailleurs.).
Before you get any replacement derailleur ordered/purchased, you’ll want to check the alignment of the derailleur hanger. This is the part where the derailleur attaches to the bike, aka where the shiny hex-head bolt on the broken piece threads into. If your hanger was twisted or tweaked when the derailleur tore off, your new unit will never shift quite right, and unfortunately it looks like your frame doesn’t have a replaceable/sacrificial hanger. You’ll either need a [hanger alignment tool](https://www.chainreactioncycles.com/us/p/lifeline-pro-gear-hanger-alignment-tool?color=silver&toolSizes=One+Size&utm_source=google&utm_medium=organic&utm_campaign=organic-shopping), or the attention of a good bike shop.
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Fuck it, go single speed!
Basically any modern derailleur is compatible with friction shifting to a point. Try to stick with sub 10 speed shimano compatible derailleurs though (as some older friction shifters don’t pull enough cable to use all gears on sram ESP or other low pull ratio derailleurs.).
Something like a Claris R2000, Sora R3000, Alivio M/T 4000 or M3100 would be perfect there. If you wanted a higher end derailleur you can still find [105-5701](https://www.universalcycles.com/shopping/product_details.php?id=53901&category=531)s in stock.
Before you get any replacement derailleur ordered/purchased, you’ll want to check the alignment of the derailleur hanger. This is the part where the derailleur attaches to the bike, aka where the shiny hex-head bolt on the broken piece threads into. If your hanger was twisted or tweaked when the derailleur tore off, your new unit will never shift quite right, and unfortunately it looks like your frame doesn’t have a replaceable/sacrificial hanger. You’ll either need a [hanger alignment tool](https://www.chainreactioncycles.com/us/p/lifeline-pro-gear-hanger-alignment-tool?color=silver&toolSizes=One+Size&utm_source=google&utm_medium=organic&utm_campaign=organic-shopping), or the attention of a good bike shop.