A plywood or cement board would be needed beneath the tile. It could be fastened through the foam or to sleepers with the foam between them.

What would the purpose be for a foil facing? Unless there is an air space next to it, it would only act as a vapor retarder.

Be sure to use the right density of extruded foam for this kind of installation.

It would help if you specified what material the tile is made of. Not so much as to the foam question, because you wouldn't use any foam under any tile. More to see what scenario you're planning.
Ron

The foam will allow enough flex to crack grout lines and even tiles.
Even cement board over foam will move enough to cause cracking.

Costco sells an in-floor heating system that appears fairly easy to install. You may want to take a look at it just in case it might work for your application:

Here is a link that might be useful: Costco Underfloor Heating System

Why would you need to tear out the slab to install radiant heating when you can just lay some of the electric heating mats?
Is this a DIY job? If so, I can tell you from experience that tiling over the mats isn't as easy as the photos make it look but it's worth the extra work in the long run, especially if you live in a cold climate.

If the budget doesn't allow for any radiant heat, then I would look at some of the Mannington or Armstrong flooring tiles as other options. They'd be much warmer than ceramic tile and with a rug on top you can probably do without the radiant heat.

Here is a link that might be useful: Flooring

Bantoniewicz, I can't tell where your from, but in my area [USDA zone 3] a tile floor in the basement is really cold in winter: I know because that's the situation of the bathroom I use every day-- getting out of that basement is the thing I'm looking forward to most at my new house.

As others have said, foam is right out below tile: even padded vinyl should be removed.

I don't blame you for not wanting carpet below grade, but I find that even a throw rug over the tile makes a huge difference in how much heat the floor pulls away from my [perpetually bare] feet.

I don't think I'd reccomend the electric mats: for me, the idea of heating the whole slab and the ground below it seems unwise for both economic and ecologic reasons.

If I were you, I'd look very closely at the 'Dri-Core' system, which gives you a thermal break from the slab. I don't know whether a foil material would be effective in the air space it provides. It's supposed to allow use of a wider range of floor products. Since I didn't do it myself, I can't tell you how well it works, and a lot depends on how you use the space.

In a cold climate, tile without rugs will pull the heat right out of bare or sockfooted feet. If its daily use space, rugs will help a lot, and be easy to remove if you do end up with a moisture issue. If its not daily use, tile without rugs or the Dricore may suit you just fine-- no flooring resists moisture better.

Here is a link that might be useful: Dricore

Thanks for all the great advice. In the end we simply tiled and will use some bath mats, etc. With all of the final decisions coming into play it all became a bit overwhelming. We do however, still have an open slab basement and are seeking to "insulate" it a bit.

Based on everyone's responses, it appears it's still possible and feasible to install a thin layer of XPS foam (e.g. R5) with a floating cork floor or FLOR carpet tiles. Based on Mightyanvil's response, we'd add a plywood baselayer (.5") between this. Any other thoughts?

You can put your foam down. There's your thermal break.

Then you can float two layers of plywood over the foam. Stagger the seams by a half a sheet in each dimension between the two layers of ply, and screw the top layer to the bottom.

Then you can apply your cork or carpet tiles to the ply.

As an alternative you could use sleepers with one layer of 3/4" CDX on top. Cork/carpet tiles over that.

Or you could do foam and a single sheet of ply, running tapcons through the ply, through the foam, and into the slab. That'd be my last choice though. Ugly work running all those blue screws.

Mongo

"Then you can float two layers of plywood over the foam. Stagger the seams by a half a sheet in each dimension between the two layers of ply, and screw the top layer to the bottom".

Sounds to unstable,imo. Even if you used 13 ply cabinet grade ply, too much risk of warping and flexing for tile. Cork or carpet tiles maybe, but i wouldn't.

I like the sleepers tapcon'd and adhesived to the slab with the foam panels inbetween, then ply glued and screwed over that.

I wrote "...cork or carpet tiles...", for clarification I should have written "cork tiles or carpet tiles".

Correct, I wouldn't apply ceramic tile over the ply. Ceramic tile would be a no go for me even if using an uncoupling membrane.

But the two layers of ply? Plenty good, half-inch over 3/4 inch, it won't walk away.

Now if he's gonna use big box store provided 3-ply 3/4" and 2-ply half-inch? It'd probably turn out as flat as a potato chip.

10-4, 13 ply would be cost prohibitve and probably overkill, but there are good 7-ply products out there. I believe the most i've seen at the big box stores in our area is 5-ply with the exception of their cabinet grade stock, which is shaky as well.

Wedi is a foam made for tile. I have used it. You tile directly onto it. You walk on the tile floor forever after.

Any other thermal break is OK too. A 1/8" or 1/4" membrane is better than none, far better. You have to separate your floor from the slab. This is not just decoupling for mechanical reasons, this is a thermal break. Otherwise you have a thermal bridge to an uninsulated slab.

I would use Wedi.

You could also lay it on top of your other product idea, the foil reflector faced foam or membrane. I would get huge tiles not small ones.

David

Interesting: never heard of this company before...

Here is a link that might be useful: Wedi products