I'm trying to figure out what you have going on back there. What you wrote seems like an area with weed fabric, mulch over that and sod over that which is nonsensical.
Something like that, yeah. No clue what the previous owners were thinking.
Is it 50% sod, 50% mulch over fabric with plants growing through or something?
Nope. It actually appears to be like one layer on top of another layer. Like the previous owner decided they didn't like what was there and instead of tearing it out and replacing it, just dropped a new layer of crap right on top and then seeded it.
Or you use plastic sheeting and cook the soil underneath while blocking any moisture from moving in or out..
They used plastic sheeting. But underneath that is a completely separate layer of stuff, including what looks like paved walking stones. Like they just rolled the sheet on top of what was there and dumped a layer of bark/soil/whatever on top of that, and then planted stuff on/through that. Yeah. Imagine my surprise. I was just trying to pull a couple errant plants from alongside the wall and realized the whole thing was like one solid layer of roots that pulled up in a solid sheet suspiciously easily (well, the one corner I was pulling on anyway, the whole mass is pretty heavy I assume). At the time I was just trying to clear the extra growth that was pushing out in front of the front door so as to make it easier for the guys doing work inside the house to get in and out, so I didn't investigate any further.