Is perfectly inelastic demand always coupled with inelastic supply? In the case of perfectly inelastic demand, the price is always infinite. Does it mean that the supply curve is also inelastic?
I assume you are talking about Price elasticity of demand/supply and not cross elasticity or income elasticity of demand/supply.

Not necessarily.

Perfectly inelastic demand means that a change in price leads to no change in quantity demanded. Imagine a country where, say, tomato is super crucial to survival, any change in its price will not make people want to buy more/less of it.

Inelastic supply can exist separately from perfectly inelastic demand. This is just how responsive supply is to changes in price. Just because the price increases/decreases, doesnt mean producers can make more/less of it more easily, even if they wanted to, e.g. think about sudden weather changes for crop production as an example.

In conclusion, no perfectly inelastic demand is not always coupled with inelastic supply, but it can be.