The singer -- who's been open about loving 'Real Housewives' -- finally got to meet Kyle during a shopping trip Thursday in Aspen ... which turned into a stroke of good luck for both of 'em.

Sources with direct knowledge tell TMZ ... Rihanna was shopping at Kemo Sabe -- after hours in a private room, because that's how she rolls -- and that's when Kyle strolled in to do her VIP shopping.

When a store employee mentioned RiRi was there, Kyle asked if they could meet ... and we're told Rihanna excitedly responded, "F*** yes!!!"

After all, our source says the whole reason Rihanna was shopping at Kemo Sabe in the first place was because she'd seen Kyle go there on 'RHOBH.'

We're told Rihanna and Kyle hugged it out and had a fun conversation ... then Mauricio Umansky and their daughters came up and met RiRi and A$AP Rocky.

Both families ended up hanging out inside the store for close to an hour ... and Kyle gifted Rihanna an $895 turquoise beaver hat.

Rihanna and A$AP had their kids in tow too as they did some big-time shopping -- we're told they bought a ton of items, including hats, boots, belts, and buckles ... plus the brown cowboy hat she wore out of the store.

Aspen's full of celebs this time of year, but it's pretty incredible when they run into each other, and they're both such big fans of each other -- like Kyle and Rihanna -- and they're not afraid to show it.