My first thought is the shiny wood floor would not flow into the not-shiny wood-look tile, so to keep the continuity I vote for the same material for both.

I vote against the "real wood" sitting beside "faux wood". It normally highlights how "fake" the faux wood looks and thereby negates the whole process.

There are ways to create flow without resorting to "look alike" products. A design professional can help with this and other design concepts. For instance, the same colour TONES can be used in two different materials...this can offer flow.

Your options included stained concrete. When worked in the same tones (probably browns) as the wood, it can offer a magnificent space that "flows" from the interior wood to the exterior slab.

Flow is a concept that is put together beginning in the design stage (if you are lucky enough to work from scratch with a designer) that "ends" when all the pieces of the puzzle are in place. Flow doesn't happen just because the same materials are used...many homes feel disjointed and yet they have all the same colours, floors, doors and trim used throughout.

I'm wondering if a design professional would be best suited to be on-site to help with this one.

I am standing with the person in this pairing who thinks trying to match real wood with wood looking tiles is a design nightmare. LOL. I voted for the wood inside and stained concrete outside in a complementary color. I can see the flow going very well depending on the color you choose. I also think concrete patios are super easy to care for.