I'd think it was a mistake but as she had to go through a gate to get to the line, she must know she's on your property.
I think it's telling that you started by calling her a neighbour rather than a tenant. Tenant- landlord relations are different from neighbour to neighbour relations. Different expectations and different power dynamic too.
As a tenant I wouldn't install a washing line without the landlords express consent. There's no way I'd assume the landlord would be ok with me driving nails into the side of their house or making a hole in the lawn.
If there were already hooks afixed to brickwork or an existing hole in the ground/ patio/ with or without cement, only then would I think it was okay to put up my own washing line.
But then, I also wouldn't be wandering onto the landlords property, leaving gates open when there are small children around (&/or animals)... Surely it's one of the absolute basics of countryside life that you keep gates shut, regardless of what and where the gate is, and what's being farmed, you just don't do it unless you want to be hung, drawn and quartered!
In the face of such rude behaviour I'd be taken aback too OP, and it's a bit mean for posters to assume you are a push over or pathetic in real life.
Surely it makes it worse that she's done it for 4 weeks and you didn't mention it the first time it happened. To be honest that's exactly the kind of situation I'd get into and then cringe over! Now you have to make a definite stand and it feels awful to have to do it. Id be cringing at drawing attention to her rudeness but I do know that's not the right way to think about it... I empathise too much and put myself into the other persons shoes, thinking 'I'd feel so embarrassed if I realised how rude my behaviour was'. But of course, you have to remember that you wouldn't have opened a garden gate to use your landlords washing line without asking first.
So my advice is:
- empathise less, she's not you, and so won't feel upset / embarrassed in the same way you would
- get comfortable with unequal power in relationships, so you don't feel terrible about enforcing that power when needed
- Also don't worry about being inconsistent, you're telling her now, and that's fine, it's not that by leaving it you lose the right to enforce your own boundaries.
I'm interested though, I wonder if she comes from somewhere where communal washing lines are the norm? Not an excuse, just me being curious!