Comment, not an answer (but it might be): [I am new and won't let me 'comment' until '50' reputation; but it DOES let me 'answer', go figure...]
Correct, it is a No-no to cross transformer supplies (at the R's). Most equipment now have one side of the 24VAC transformer physically attached to ground. This will lead to having both (or potentially 3, if AC has it's own) 'crossed' in parallel. Both transformers could get hot if not precisely the same voltage, and would actually be a 48V short, if their outputs happen to be 180° out of phase; 50/50 chance!). And not safe in case of a future replacement of a transformer, or unit. I believe it should be isolated for serviceability in the future.
My opinion is one system should be put on a 24VAC relay. I would put the primary (hydronic) control on the relay output contacts (isolating that one's transformer), and run the thermostat and the furnace on the furnaces transformer. (this way future AC changout would not present any issues, and the R and Rc feature on the thermostat can be split for any AC needing that).
I think any standard small 24VAC relay with say 3A contacts will be adequate. The hydronic W probably draws less that 1.6A, possibly near zero.
To wire the relay so the coil is on the furnace 'side' and the contacts are on the hydronic side, would go like this:
The relay will need a connection for the 'ground-side' of the coil, probably the 'blue' transformer wire of the furnace transformer. The furnace C is probably available at your thermostat too, (blue?) depending on where you choose to place the relay.
The thermostat W1 would go to the coil 'hot'. The contacts are wired to the 'red to white' hydronic cable, to 'make' the connection for the hydronic W signal.
Since the thermostat has an LCD display, it likely has both 24VAC supply wires at the thermostat location, and won't need to run an added wire. You will need 2-wire red-white to go to the hydronic, in case there is only one white there now.
The R from your hydronic would normally go to your thermostat R, but now will go to one of the relay contacts. The hydronic W would normally go to your thermostat W, or W1 on dual stage; but now will go to the other relay contact.
The relay coil is now going to be connected to the thermostat W, and the 'ground' for the coil will be connected to the C wire (blue?) from your FURNACE, NOT the hydronic. Relay is now a 'go-between', receiving the output of the thermostat W, and closing a pair of contacts that are connect to the cable from your hydronic.
Connect R-Rh with jumper, and the furnace is powering everything now, (except the hydronics is running on it's own transformer).
Coil is powered from the furnace side; contacts connect only to the hydronic side.
I think I described that correctly. Any mistakes please correct me, anyone. This is how I would do it, if I was doing it. I can be more specific if you need, but would need more details of your layout.
Mr K L AC technician
Ok, I got your system/situation figured out; as to why your furnce is currently not functioning correctly:
You must have the Rc-Rh jumper removed (a good thing), and your hydronic (or your furnace) does not have a ground on it's transformer. Apparently one or both of the transformers are direct wired to the controls, without using a frame ground (providing a 'return' path). [This is only a problem when one unit is trying to activate something 'outside' of itself]. Or (if you find both transformers are grounded, then) one of the units itself is not grounded good, or not grounded at all.
So the Rh from the hydronic's transformer has no voltage (to 'ground') relative to the furnace's circuitry. The furnace transformer is powering the AC fine, but the thermostat is using the hydronic's transformer to power the hydronic's heat (works fine), AND the furnace's heat (which has no path for the 'return' current back to the transformer).
You can confirm this by checking the AC voltage from Rh (hydronics power) to C from the furnace. (this may be different when 'loaded', ie it may show 24VAC until W2 demand kicks in and tries to draw actual current).
I think you could theoretically do this without a relay, but you will need the 'ground' side of both transformers tied. Check if each unit has one side of it's transformer grounded to the frame, and if the 120V ground to the building is intact/present. A (blue?) wire from one unit to the other would insure direct current return, instead of depending on earth ground (and it is not proper to use ground to conduct current, even low voltage AFAIK).
I also don't know if the hydronics transformer is rated to also power additional loads like this, or if the 48VAC issue can damage these controller boards. The relay is probably the only right way to do this. It also makes sure that ANY future replacement of either unit (or the thermostat) will not have to be customized to work right... If you use a relay, you won't have to worry about the grounds, or a return path between units. (It makes both units' power circuits totally independent).
First, I would test your furnace alone to insure nothing has been damaged, and then try the relay setup.
Set it up the normal way: Temporarily unhook the hydronics, put the furnace on W1, jumper Rh-Rc, and test it. I am confident it will work normally again. [it's the transformers' grounding and/or lack of return path that prevented the W2 from functioning].
Oh, With the relay you will have Rh-Rc jumpered, the furnace transformer will power the furnace stage 2 heat, the thermostat, the relay, and the AC cycle. The hydronics transformer will only power itself, (totally isolated), and is only controlled by the relay.
Mr K L
Your post was clear, my head was not. LOL I misunderstood, and missed some things that were clearly stated in your post. (also while in 'edit post' mode, I can't see the OP to refer to)