Yes they can. It's not required for there to be a turn-based action in a given phase or step for people to get priority. Upkeep, main phase, beginning of combat, end of combat and end step all are steps and phases that have no turn-based action, but where people get priority. In any step or phase where people get priority, people can cast spells and activate abilities (that's what having priority means), people all get priority before any spell or non-mana ability resolves (like in your example, before Tidal Warrior's ability resolves), and people all get priority before that step or phase can end on an empty stack (in your example, your opponent can even wait for Tidal Warrior's ability to resolve to cast Entomb, as long as they put black mana in their mana pool before their black source becomes an Island.)
Turn-based actions are things like drawing a card as the first action taken by the active player during his draw step, declaring attackers, declaring blockers (and then the controller of attackers deciding damage assignment order), assigning and dealing combat damage, discarding to maximum hand size in cleanup step.
I'm a former judge (lapsed), who keeps up to date on rules and policy. Keep in mind that judges' answers aren't necessarily more valid than those of people who aren't judges; what matters is we can quote the rules to back up our answers. When in doubt, ask for such quotes.
So if they didn't have an instant in that situation, can they just tap the swamp in response and float B?
Yes, but mana empties from the mana pool as any step or phase ends, so if they have nothing to spend that B on in that same upkeep, that'll be in vain.
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I'm a former judge (lapsed), who keeps up to date on rules and policy. Keep in mind that judges' answers aren't necessarily more valid than those of people who aren't judges; what matters is we can quote the rules to back up our answers. When in doubt, ask for such quotes.
A related point: If you could use your creature's ability, which can be activated any time anyone could cast an instant... then why couldn't they cast an instant?
Another note: Turn-based actions have nothing to do with anything here. Turn-based actions are:
"703.1. Turn-based actions are game actions that happen automatically when certain steps or phases begin, or when each step and phase ends. Turn-based actions don’t use the stack."
That is, things like untapping in the untap step, drawing in the draw step, the various actions taken during combat, etc. They are not controlled by players -- they just sorta happen. Casting spells or activating abilities are, as MadMage explained, very different things.
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"It is better for all the world if, instead of waiting to execute degenerate offspring for crime or to let them starve for their imbecility, society can prevent those who are manifestly unfit from continuing their kind. The principle that sustains compulsory vaccination is broad enough to cover cutting the Fallopian tubes... Three generations of imbeciles are enough."
--Buck v Bell, 1927. This case, regarding the compulsory sterilization of inmates at mental institutions, has -- somehow -- never been overturned. Just a wee PSA for ya.
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At the beginning of upkeep on opponent's turn, I tap Tidal Warrior to turn a swamp into an island.
Can that person tap that swamp and use it as mana to play an instant, like Entomb?
A rule stated, 503.1. The upkeep step has no turn-based actions...
But there was some discussion back and forth about what happens there.
Turn-based actions are things like drawing a card as the first action taken by the active player during his draw step, declaring attackers, declaring blockers (and then the controller of attackers deciding damage assignment order), assigning and dealing combat damage, discarding to maximum hand size in cleanup step.
So if they didn't have an instant in that situation, can they just tap the swamp in response and float B?
Another note: Turn-based actions have nothing to do with anything here. Turn-based actions are:
"703.1. Turn-based actions are game actions that happen automatically when certain steps or phases begin, or when each step and phase ends. Turn-based actions don’t use the stack."
That is, things like untapping in the untap step, drawing in the draw step, the various actions taken during combat, etc. They are not controlled by players -- they just sorta happen. Casting spells or activating abilities are, as MadMage explained, very different things.
--Buck v Bell, 1927. This case, regarding the compulsory sterilization of inmates at mental institutions, has -- somehow -- never been overturned. Just a wee PSA for ya.