Kyle Richards Supports Estranged Husband Mauricio Umansky for 'DWTS' Debut

Kyle Richards and Mauricio Umansky's separation is the nicest one in Hollywood history -- 'cause the guy's estranged wife showed up in person to watch him dance on live TV.

The 'RHOBH' star arrived for Tuesday's taping of "Dancing With the Stars" -- which debuted its latest season on ABC ... showing off all the contestants and their round-1 moves, including Mauricio and his dance partner Emma Slater's jive number.

His first outing wasn't all that great numbers-wise -- he scored just a 15/30 from the judges -- but either way ... Kyle was out there to see him do his thing, sitting front and center and loudly cheering MU on. She also brought out their two daughters for extra support.

Obviously, this is interesting ... y'know, considering they're currently "separated" and all -- even though they're clearly on good terms and continue to live in the same home all kumbaya.

Mauricio's little pre-taped introduction video actually touched on his relationship with Kyle ... he said it'd been a rough year and that he was hoping 'DWTS' could serve as a good distraction.

Remember ... the 'DWTS' folks actually wanted not just Mauricio on this season, but they actually made a play to get Kyle and her good female pal Morgan Wade as well -- but that didn't materialize in the end.

Kyle and Morgan's close relationship has led some to suspect they might be more than friends -- but we've been repeatedly told they're just friends and business associates. Kyle is working on a documentary about Morgan's life/career.

As for Mauricio ... he's already told us he's cool with Morgan -- and he clearly remains cool with Kyle too, as they continue to spend holidays together with the whole family.