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The Allied player controls the Ionian Sea with a carrier and a plane, blocking supply from Rome.
That turn, I use the contingency 'Bulgaria signs Pact', and deploy an infantry in Bulgaria to go with my tank already there. The card says I should also now be able to supply from Bulgaria.
However, when the turn ends both my units are unsupplied.
Have I made an error or is this actually a bug?

Seems like a bug after you declared war on Bulgaria first (move 187) instead of activating it through the card, which there should never be a need to. Though in the rules it doesn't look like declaring war on Bulgaria should stop the supply marker being placed.

I think you're misunderstanding the first part of the card, "Activate Bulgaria for the Axis if possible". This has Bulgaria and it's Minor Force Counter join your side so there's no need to waste a card to attack it.

It also leaves the Bulgarian army there so you can turn it into a pact Infantry with the flip side of that contingency card.