As part of a basement finishing job, I'm planning on putting down 1" XPS to warm up the floor prior to the finished floor. I plan to use plywood on top of the XPS to support either laminate or engineered wood. However, I'd like to tile over some of that subfloor for a bathroom and walkout entry way and I'm wondering if the XPS is rigid enough to support a tile installation? I could imagine 1"XPS then cement board, or plywood and ditra. I've seen a commercial product made by amdry that seems to mimic this concept, but it is pretty expensive. Has anyone done anything like this?

You can get XPS with different compression strengths. Sounds like you are sure the floor will stay dry - consider polyiso (more R value in the 1" and better for the environment).

I don't know the answer, but a 12x12" tile is 144 sq inches and at 20 psi, that's nominally over a ton, without even considering the effect of the cement board or grout. 1" square tiles would be quite different (but perhaps fine too).

Fully supported by a slab and a 3/4" subfloorabove it you don't have to think AT ALL about the compressive strength of the foam. The subfloor on a foam/slab sandwich flexes far less than if it were being supported by joists 16" on center. The key is to stagger the seams of the subfloor with those of the floor- the most compressible part of the foam will be at the seams, and the most flexible part of the subfloor are the seams. As long as there's a foot or so of overlap, it won't develop a problem over time.

Polyiso on a slab is a risky proposition, but EPS would be fine, and considerably greener (due to the differences in blowing agents used- it's the same polymer.) The performance of XPS eventually falls to that of EPS of equal density over time (mind you, it's a fairly long time, maybe 50 years). It takes 1.2" of EPS to equal the same labeled performance of 1" XPS.

Fair point on the green-ness of EPS over XPS.

An interesting question would be "if you use 12x12 tiles, can you use 1.5" foam and skip the cement board or plywood?". 1/4" Hardibacker might be another option that would allow more R value.

Another thing to consider would be foam-core backer board. The options include Wedi or USG Ultralight which are similar fiberglass over XPS foam, made in different thicknesses. Wedi uses european CO2-blown XPS so it's got low global warming impact, whereas as best I can tell, USG uses american HFC blown XPS for high global warming impact. With those, you could just buy the total thickness you want and need nothing but that and the tile. There are also 1/2" foam core boards from JM (GoBoard) and FinPan (ProPanel). JM uses a polyiso core and FinPan a high-density EPS core. JM claims GoBoard is fully waterproof, but I'd feel a little safer about the moisture performance of FinPan. But with those, you'd have to either stack up multiple layers, or use it in addition or instead of the plywood layer on top of the foam.

All in all Wedi sounds to me like a good solution for this.